1 Introduction
Let p be a rational prime and let N be a positive integer which is prime to p. We fix an algebraic closure
$\bar {\mathbb {Q}}_{p}$
of
$\mathbb {Q}_{p}$
and denote its p-adic completion by
$\mathbb {C}_{p}$
. Let
$\mathcal {W}_{\mathbb {Q}}$
be the weight space for
$GL_{2,\mathbb {Q}}$
, which is a rigid analytic variety over
$\mathbb {Q}_{p}$
such that the set of
$ \mathbb {C}_{p}$
-valued points
$\mathcal {W}_{\mathbb {Q}}\left (\mathbb {C}_{p}\right )$
is identified with the set of continuous homomorphisms
$\mathbb {Z}_{p}^{\times }\to \mathbb {C}_{p}^{\times }$
.
In [Reference Coleman and Mazur15, Reference Buzzard12], Coleman and Mazur and Buzzard defined a rigid analytic curve
$\mathcal {C}_{N}$
with a morphism
$\kappa : \mathcal {C}_{N}\to \mathcal {W}_{\mathbb {Q}}$
such that the set of
$\mathbb {C}_{p}$
-valued points
$\mathcal {C}_{N}\left (\mathbb {C}_{p}\right )$
is in bijection with the set of normalized overconvergent elliptic eigenforms of tame level N which are of finite slopes, in such a way that the eigenform f corresponding to a point
$x\in \mathcal {C}_{N}\left (\mathbb {C}_{p}\right )$
is of weight
$\kappa (x)$
. The curve
$\mathcal {C}_{N}$
is called the Coleman–Mazur eigencurve, and it has played an important role in arithmetic geometry, since it turned out to be useful to control p-adic congruences of elliptic modular forms. After their construction of the eigencurve, much progress has been made to generalize it to the case of automorphic forms on algebraic groups other than
$GL_{2,\mathbb {Q}}$
. Now we have, for various algebraic groups G over a number field, a similar rigid analytic variety
$\mathcal {E}$
to the Coleman–Mazur eigencurve over a weight space
$\mathcal {W}^{G}$
for G, which is called the eigenvariety for G.
Despite their importance, we still do not know much about the geometry of eigenvarieties. For example, we do not even know if an eigenvariety has finitely many irreducible components. One of the topics of active research is the smoothness of eigenvarieties at classical points. For the Coleman–Mazur eigencurve, we know the smoothness at classical points in many cases [Reference Bellaïche and Chenevier6, Reference Bellaïche and Dimitrov8, Reference Hida30, Reference Kisin32]. Bellaïche and Chenevier [Reference Bellaïche and Chenevier7] studied tangent spaces of their eigenvariety for unitary groups at certain classical points, and applied it to showing the nonvanishing of a Bloch–Kato Selmer group. On the other hand, Bellaïche proved the nonsmoothness of the eigenvariety for
$U(3)$
at classical points [Reference Bellaïche5]. It is natural to think that such geometric information of eigenvarieties is related to deep p-adic properties of automorphic forms.
Another interesting topic, which this paper is concerned with, is the properness of eigenvarieties over weight spaces. Since eigenvarieties are not of finite type over weight spaces, they are not proper in the usual sense. Instead, we consider the following geometric interpretation of the nonexistence of holes: Let
$ \mathcal {D}_{\mathbb {C}_{p}}=\mathrm {Sp}\left (\mathbb {C}_{p}\langle T \rangle \right )$
be the closed unit disc centered at the origin O and let
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}=\mathcal {D}_{\mathbb {C}_{p}}\setminus \{O\}$
be the punctured disc. For any quasi-separated rigid analytic variety
$\mathcal {X}$
, we write
$\mathcal {X}_{\mathbb {C}_{p}}$
for the base extension of
$\mathcal {X}$
to
$\mathrm {Sp}\left (\mathbb {C}_{p}\right )$
. Suppose that we have a commutative diagram of rigid analytic varieties
where the vertical arrows are the natural maps. Then we say that the eigenvariety
$\mathcal {E}$
is proper if there exists a morphism
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {E}_{\mathbb {C}_{p}}$
such that the diagram is still commutative with this morphism added. Roughly speaking, this means that any family of overconvergent eigenforms of finite slopes on G parametrized by the punctured disc can always be extended to the puncture. However, note that what eigenvarieties parametrize are in general not eigenforms themselves but eigensystems occurring in the space of overconvergent automorphic forms. We also note that the naive interpretation of the nonexistence of holes – that any p-adically convergent sequence of overconvergent eigenforms of finite slopes converges to an overconvergent eigenform of finite slope – is false [Reference Coleman and Stein16, Theorem 2.1].
For the properness of the Coleman–Mazur eigencurve
$\mathcal {C}_{N}$
, Buzzard and Calegari first proved the properness of
$\mathcal {C}_{N}$
for the case where
$p=2$
and
$N=1$
[Reference Buzzard and Calegari13]. It was followed by Calegari’s result [Reference Calegari14] on the properness of
$\mathcal {C}_{N}$
at integral weights: he showed the existence of the map
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {C}_{N,{\mathbb {C}_{p}}}$
as in the definition of properness if the image of the puncture O in the weight space corresponds to a classical weight. One of the key points of their proofs is to show that any nonzero overconvergent elliptic eigenform of infinite slope does not converge on a certain region of a modular curve, while any overconvergent elliptic eigenform of finite slope does converge on a larger region. In [Reference Buzzard and Calegari13], they deduced the former from the theory of canonical subgroups, especially a behavior of the
$U_{p}$
-correspondence for elliptic curves with Hodge height
$p/(p+1)$
, and the latter is a consequence of a standard analytic continuation argument via the
$U_{p}$
-operator. Recently, the properness of the Coleman–Mazur eigencurve was proved in full generality by Diao and Liu [Reference Diao and Liu22] using p-adic Hodge theory, especially the theory of trianguline p-adic representations in families.
For algebraic groups other than
$GL_{2,\mathbb {Q}}$
, the properness of eigenvarieties has not been known. Note that in Diao and Liu’s proof of the properness of the Coleman–Mazur eigencurve, in order to apply p-adic Hodge theory it seems crucial that we have a Galois representation, not just a Galois pseudorepresentation, over (the normalization of) the eigencurve. This is a consequence of the fact that we can convert pseudorepresentations into representations over smooth rigid analytic curves [Reference Coleman and Mazur15, remark after Theorem 5.1.2]. Thus, at present, it is unclear if their proof can be generalized to show the properness of eigenvarieties of dimension greater than
$1$
on the components where the residual Galois representations attached to automorphic forms are absolutely reducible.
The aim of this paper is to generalize the method of Buzzard and Calegari to the case of Hilbert modular forms and to obtain the properness of the Hilbert eigenvariety constructed by Andreatta, Iovita, and Pilloni [Reference Andreatta, Iovita and Pilloni4] at integral weights in some cases.
To state the main theorem, we fix some notation. For any totally real number field F with ring of integers
$\mathcal {O}_{F}$
, put
$G=\mathrm {Res}_{F/\mathbb {Q}}(\mathit {GL}_{2})$
and
$\mathbb {T}=\mathrm {Res}_{\mathcal {O}_{F}/\mathbb {Z}}(\mathbb {G}_{\mathrm {m}})$
. Let
$K/\mathbb {Q}_{p}$
be a finite extension such that
$F\otimes K$
splits completely. Let
$\mathcal {W}^{G}$
be the weight space for G over K as in [Reference Andreatta, Iovita and Pilloni4, §4.1]. By definition, we have
$$ \begin{align*} \mathcal{W}^{G}=\mathrm{Spf}\left(\mathcal{O}_{K}\left[\left[\mathbb{T}\left(\mathbb{Z}_{p}\right)\times \mathbb{Z}_{p}^{\times}\right]\right]\right)^{\mathrm{rig}}, \end{align*} $$
and the set of
$\mathbb {C}_{p}$
-valued points
$\mathcal {W}^{G}\left (\mathbb {C}_{p}\right )$
can be identified with the set of pairs of continuous characters
$$ \begin{align*} \nu:\mathbb{T}\left(\mathbb{Z}_{p}\right)\to \mathbb{C}_{p}^{\times},\qquad w:\mathbb{Z}_{p}^{\times}\to \mathbb{C}_{p}^{\times}. \end{align*} $$
We say that the weight
$(\nu ,w)$
is
$1$
-integral if its restriction to
$1+p\left (\mathcal {O}_{F}\otimes \mathbb {Z}_{p}\right )\times \left (1+p\mathbb {Z}_{p}\right )$
comes from an algebraic character
$\mathbb {T}\times \mathbb {G}_{\mathrm {m}} \to \mathbb {G}_{\mathrm {m}}$
. This restriction corresponds to a pair
$\left (\left (k_{\beta }\right )_{\beta },k_{0}\right )$
of a tuple
$\left (k_{\beta }\right )_{\beta }$
of integers indexed by the set of embeddings
$\beta : F\to K$
and an integer
$k_{0}$
. We say that a
$1$
-integral weight is
$1$
-doubly even if every
$k_{\beta }$
and
$k_{0}$
are divisible by
$4$
. Then the main theorem in this paper is the following:
Theorem 1.1 (Theorem 5.1)
Let F be a totally real number field which is unramified over p. Let
$K/\mathbb {Q}_{p}$
be a finite extension in
$\bar {\mathbb {Q}}_{p}$
such that
$F\otimes K$
splits completely. Let
$N\geq 4$
be an integer prime to p. Let
$\mathcal {E}\to \mathcal {W}^{G}$
be the Hilbert eigenvariety of tame level N constructed in [Reference Andreatta, Iovita and Pilloni4, §5], which is defined over a finite extension of K.
Suppose that for any prime ideal
$\mathfrak {p}$
of F dividing p, the residue degree
$f_{\mathfrak {p}}$
of
$\mathfrak {p}$
satisfies
$f_{\mathfrak {p}}\leq 2$
(resp., p splits completely in F) if p is odd (resp., even). Then
$\mathcal {E}$
is proper at
$1$
-integral (resp.,
$1$
-doubly even) weights. Namely, any commutative diagram of rigid analytic varieties over
$\mathbb {C}_{p}$
can be filled with the dotted arrow if
$\psi (O)$
corresponds to a
$1$
-integral (resp.,
$1$
-doubly even) weight.
For the proof, we basically follow the idea of Buzzard and Calegari [Reference Buzzard and Calegari13, Reference Calegari14]. Thus the key step in our case is also to show that any nonzero overconvergent Hilbert eigenform f of
$1$
-integral weight and infinite slope does not converge on the locus where all the partial Hodge heights are no more than
$1/(p+1)$
in a Hilbert modular variety.
Let us explain briefly how to show this nonconvergence property, following [Reference Buzzard and Calegari13]. For simplicity, we assume that f is of integral weight, namely the weight
$(\nu ,w)$
corresponds to an algebraic character
$\mathbb {T}\times \mathbb {G}_{\mathrm {m}}\to \mathbb {G}_{\mathrm {m}}$
. For any Hilbert–Blumenthal abelian variety (HBAV) A with an
$\mathcal {O}_{F}$
-action over the ring of integers
$\mathcal {O}_{L}$
of a finite extension
$L/K$
, we say that a finite flat closed
$\mathcal {O}_{F}$
-subgroup scheme
$\mathcal {H}$
of A over
$\mathcal {O}_{L}$
is p-cyclic if its generic fiber is étale locally isomorphic to the constant group scheme
$\underline {\mathcal {O}_{F}/p\mathcal {O}_{F}}$
. We say that A is critical if every
$\beta $
-Hodge height of A is equal to
$p/(p+1)$
for any embedding
$\beta : F\to K$
. Then, for any critical A and any p-cyclic subgroup scheme
$\mathcal {H}$
of A, the quotient
$A/\mathcal {H}$
has the canonical subgroup
$A[p]/\mathcal {H}$
of level
$1$
and its
$\beta $
-Hodge heights are all
$1/(p+1)$
[Reference Hattori29, Proposition 6.1]. This is where the assumption on residue degrees is used in the most crucial way. It is unclear if the claim holds without this assumption: at least, we have a counterexample of a similar assertion for truncated Barsotti–Tate groups if we drop the assumption on
$f_{\mathfrak {p}}$
[Reference Hattori29, Remark 6.2].
Consider the Hilbert modular variety classifying pairs
$(A,\mathcal {H})$
of an HBAV A and its p-cyclic subgroup scheme
$\mathcal {H}$
. Let
$\mathcal {U}$
be the locus where
$\mathcal {H}$
is the canonical subgroup of A. Another thing we need here is to show that for any
$(A,\mathcal {H})$
with A critical, the corresponding point
$[(A,\mathcal {H})]$
of the Hilbert modular variety has a connected admissible affinoid open neighborhood intersecting
$\mathcal {U}$
such that, if an overconvergent Hilbert eigenform f of integral weight converges on the locus where all the
$\beta $
-Hodge heights are
$\leq 1/(p+1)$
, then we can evaluate
$U_{p}f$
on this neighborhood (Proposition 3.7). This implies that if f is in addition of infinite slope, then we have
$\left (U_{p}f\right )(A,\mathcal {H})=0$
for any critical A and any p-cyclic subgroup scheme
$\mathcal {H}$
. From this, by a combinatorial argument (Lemma 5.2), we obtain
$f(A/\mathcal {H},A[p]/\mathcal {H})=0$
for any such
$(A,\mathcal {H})$
, which yields
$f=0$
, and the nonconvergence property follows. It seems that this argument using a connected neighborhood cannot be generalized immediately to the case where f is not of locally algebraic weight, since in this case
$U_{p} f$
is defined only on the locus
$\mathcal {U}$
(even after taking a finite étale cover) and it cannot be evaluated for any critical A.
Note that sheaves of overconvergent Hilbert modular forms of [Reference Andreatta, Iovita and Pilloni4] are defined on the locus in the Hilbert modular variety where canonical subgroups exist. However, the theory of canonical subgroups used in [Reference Andreatta, Iovita and Pilloni4] does not give the existence locus which is enough large to contain critical HBAVs unless p is sufficiently large. Instead, we use [Reference Hattori29, Theorem 8.1], which enables us to enlarge the locus where sheaves of overconvergent Hilbert modular forms are defined from the original locus given in [Reference Andreatta, Iovita and Pilloni4], and to include the case of
$p<5$
in the main theorem.
What the Hilbert eigenvariety
$\mathcal {E}$
of [Reference Andreatta, Iovita and Pilloni4] parametrizes are eigensystems in the space of overconvergent Hilbert modular forms. Thus, to follow the strategy of Buzzard and Calegari to reduce properness to the nonconvergence property of overconvergent modular forms, we have to convert a family of eigensystems of finite slopes, or a morphism from a rigid analytic variety to
$\mathcal {E}$
, into a family of eigenforms and vice versa. The latter direction can be treated (Proposition 2.7) as in the proof of [Reference Bellaïche and Chenevier7, Proposition 7.2.8]. For the former direction, we first prove that any family of eigensystems over any smooth rigid analytic variety over
$\mathbb {C}_{p}$
can be lifted locally to a family of eigenforms (Proposition 2.5). This can be considered as a version of Deligne and Serre’s lifting lemma [Reference Deligne and Serre21, Lemme 6.11]. Then we glue the local eigenforms using a weak multiplicity one result, after we normalize the local eigenforms with respect to the first q-expansion coefficient (Proposition 4.15). This use of weak multiplicity one and normalization via a q-expansion coefficient hinders us from generalizing the main theorem to the case of
$GSp_{2g}$
, where sheaves of overconvergent Siegel modular forms and the Siegel eigenvariety are constructed in a similar way [Reference Andreatta, Iovita and Pilloni3].
Once we have a family of overconvergent Hilbert eigenforms f of finite slopes parametrized by
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}$
associated to the family of eigensystems
$\varphi :\mathcal {D}^{\times }_{\mathbb {C}_{p}} \to \mathcal {E}_{\mathbb {C}_{p}}$
, we extend its domain of definition in the Hilbert modular variety as large as possible by an analytic continuation using the
$U_{p}$
-operator. Since the Hecke eigenvalues are of absolute values bounded by
$1$
, we can show that q-expansion defines a rigid analytic function around a cusp parametrized by
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}$
which is of absolute value bounded by
$1$
. Such a function automatically extends to the puncture, and a gluing shows that f also extends to the puncture (Proposition 4.19). Since we analytically continued f to a large region, the specialization
$f(O)$
at the puncture is also defined over the same large region. Thus the nonconvergence property of eigenforms of infinite slope implies that
$f(O)$
is also of finite slope, which gives us an extended map
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {E}_{\mathbb {C}_{p}}$
.
The organization of this paper is as follows. In §2, we recall Buzzard’s eigenvariety machine [Reference Buzzard12], on which the construction of the Hilbert eigenvariety in [Reference Andreatta, Iovita and Pilloni4] relies, and we prove results to convert a family of eigensystems into local eigenforms and vice versa. In §3, we recall the definition of overconvergent Hilbert modular forms and the construction of the Hilbert eigenvariety, both due to Andreatta, Iovita, and Pilloni [Reference Andreatta, Iovita and Pilloni4], including generalizations of some of their results to the case over
$\mathbb {C}_{p}$
. We also give a connected neighborhood of any critical point in a Hilbert modular variety, which is one of the key ingredients of the proof of Theorem 1.1. In §4, we prove properties of the q-expansion for overconvergent Hilbert modular forms. These are used to produce a global eigenform by gluing local eigenforms obtained from a family of eigensystems, and also to extend a family of overconvergent Hilbert eigenforms over the punctured unit disc to the puncture. Combining these results, we prove Theorem 1.1 in §5.
2 Lemmata on Buzzard’s eigenvariety
Let p be a rational prime and let K be a finite extension of
$\mathbb {Q}_{p}$
in
$\bar {\mathbb {Q}}_{p}$
. In this section, we establish two lemmata on Buzzard’s eigenvariety machine [Reference Buzzard12]. In the first lemma, we show that any family of Hecke eigensystems over a smooth rigid analytic variety over
$\mathbb {C}_{p}$
lifts locally to a family of eigenforms. The second one enables us to convert any family of Hecke eigensystems of finite slopes over a reduced rigid analytic variety into a morphism to the eigenvariety.
2.1 Buzzard’s eigenvariety machine
First we briefly recall the construction of Buzzard’s eigenvariety. Let R be a reduced K-affinoid algebra. Let M be a Banach R-module satisfying the condition
$(Pr)$
of [Reference Buzzard12, §2]. We write
$\mathrm {End}_{R}^{\mathrm {cont}}(M)$
for the R-algebra of continuous R-endomorphisms of M. Let
$\mathbf {T}$
be a commutative K-algebra endowed with a K-algebra homomorphism
$\mathbf {T}\to \mathrm {End}_{R}^{\mathrm {cont}}(M)$
. Let
$\phi $
be an element of
$\mathbf {T}$
. Suppose that
$\phi $
acts on M as a compact operator. We call such a quadruple
$(R,M,\mathbf {T},\phi )$
an input datum for the eigenvariety machine over K.
For such M and
$M^{\prime }$
, a continuous R-linear
$\mathbf {T}$
-module homomorphism
$\alpha :M^{\prime }\to M$
is called a primitive link if there exists a compact R-linear
$\mathbf {T}$
-module homomorphism
$c: M\to M^{\prime }$
such that
$\phi $
acts on M as
$\alpha \circ c$
and it acts on
$M^{\prime }$
as
$c\circ \alpha $
. A continuous R-linear
$\mathbf {T}$
-module homomorphism
$\alpha :M^{\prime }\to M$
is called a link if it is the composite of a finite number of primitive links.
Let
$P(T)=1+\sum _{n\geq 1} c_{n} T^{n}$
be the characteristic power series of
$\phi $
acting on M, which is an element of the ring
$R\{\{T\}\}$
of entire functions over R. The spectral variety
$Z_{\phi }$
for
$\phi $
is the closed analytic subvariety of
$\mathrm {Sp}(R)\times \mathbb {A}^{1}$
defined by
$P(T)$
. We denote the projection
$Z_{\phi }\to \mathrm {Sp}(R)$
by f.
The eigenvariety
$\mathcal {E}$
associated to
$(R,M,\mathbf {T},\phi )$
is the rigid analytic variety over
$Z_{\phi }$
, defined as follows: let
$\mathcal {C}$
be the set of admissible affinoid open subsets Y of
$Z_{\phi }$
satisfying the condition that there exists an affinoid subdomain X of
$\mathrm {Sp}(R)$
such that
$Y\subseteq f^{-1}(X)$
and the map
$Y\to X$
induced by f is finite and surjective. We can show that
$\mathcal {C}$
is an admissible covering of
$Z_{\phi }$
[Reference Buzzard12, §4, Theorem], and we refer to
$\mathcal {C}$
as the canonical admissible covering of
$Z_{\phi }$
.
Let
$Y=\mathrm {Sp}(B)$
be an element of
$\mathcal {C}$
and let
$X=\mathrm {Sp}(A)$
be as before. Suppose that X is connected. Then the A-algebra B is projective of constant rank d. In the ring of entire functions
$A\{\{T\}\}$
over A, we can show that
$P(T)$
can be written as
$P(T)=Q(T)S(T)$
with some
$S(T)\in A\{\{T\}\}$
and a polynomial
$Q(T)$
of degree d over A with constant term
$1$
, and that we have a natural isomorphism
$A[T]/(Q(T))\simeq B$
. Put
$Q^{*}(T)=T^{d} Q\left (T^{-1}\right )$
. By the Riesz theory [Reference Buzzard12, Theorem 3.3], the restriction
$M_{A}$
of M to
$X=\mathrm {Sp}(A)$
can be uniquely decomposed as
$M_{A}=N\oplus F$
, where N is a projective A-module of rank d such that
$Q^{*}(\phi )$
acts on N as the zero map and it acts on F as an isomorphism. Since
$Q^{*}(0)\neq 0$
, the operator
$\phi $
is invertible on N. Let
$\mathbf {T}(Y)$
be the A-subalgebra of
$\mathrm {End}_{A}^{\mathrm {cont}}(N)$
generated by the image of
$\mathbf {T}$
. Then the A-algebra
$ \mathbf {T}(Y)$
is finite and thus a K-affinoid algebra. Moreover, we have a natural A-algebra homomorphism
$A[T]/(Q(T))\simeq B\to \mathbf {T}(Y)$
sending T to
$(\phi \rvert _{N})^{-1}$
. Put
$\mathcal {E}(Y)=\mathrm {Sp}(\mathbf {T}(Y))$
. If X is not connected, by decomposing X into connected components as
$X=\coprod _{i} X_{i}$
we put
$ \mathcal {E}(Y)=\coprod _{i} \mathcal {E}\left (Y\rvert _{X_{i}}\right )$
. Then these local pieces can be glued along the admissible covering
$\mathcal {C}$
and define the eigenvariety
$\mathcal {E}\to Z_{\phi }$
[Reference Buzzard12, §5]. By [Reference Buzzard12, Lemma 5.3], the rigid analytic varieties
$\mathcal {E}$
and
$Z_{\phi }$
are separated.
By the construction, the natural map
$\mathcal {E}\to Z_{\phi }$
is finite and the structure morphism
$\mathcal {E}\to \mathrm {Sp}(R)$
is locally (with respect to both the source and the target) finite. Moreover, we have a K-algebra homomorphism
$\mathbf {T}\to \mathcal {O}(\mathcal {E})$
such that, for any admissible affinoid open subset V of
$Z_{\phi }$
, the induced map
$\mathbf {T}\otimes _{K} \mathcal {O}(V)\to \mathcal {O}(\mathcal {E}\rvert _{V})$
is surjective.
In some cases we can glue this construction to define the eigenvariety over a nonaffinoid base space. Let
$\mathcal {W}$
be a reduced rigid analytic variety over K. Let
$\mathbf {T}$
be a commutative K-algebra and let
$\phi $
be an element of
$\mathbf {T}$
. Suppose that for any admissible affinoid open subset
$X\subseteq \mathcal {W}$
, we are given a Banach
$\mathcal {O}(X)$
-module
$M_{X}$
satisfying condition
$(Pr)$
with a K-algebra homomorphism
$\mathbf {T}\to \mathrm {End}_{\mathcal {O}(X)}^{\mathrm {cont}}(M_{X})$
such that the image of
$\phi $
is a compact operator. Suppose also that for any admissible affinoid open subsets
$X_{1}\subseteq X_{2}\subseteq \mathcal {W}$
, we have a continuous
$\mathcal {O}(X_{1})$
-module homomorphism
$\alpha : M_{X_{1}}\to M_{X_{2}}\hat {\otimes }_{\mathcal {O}\left (X_{2}\right )} \mathcal {O}(X_{1})$
, which is a link and satisfies a cocycle condition. Then the eigenvarieties for
$(\mathcal {O}(X), M_{X}, \mathbf {T}, \phi )$
can be patched into the eigenvariety
$\mathcal {E}\to Z_{\phi } \to \mathcal {W}$
[Reference Buzzard12, Construction 5.7], where
$Z_{\phi }$
denotes the spectral variety over
$\mathcal {W}$
constructed by gluing the spectral varieties over X.
Let
$L/K$
be an extension of complete valuation fields (of height
$1$
). For any quasi-separated rigid analytic variety
$\mathcal {X}$
over K and any coherent
$\mathcal {O}_{\mathcal {X}}$
-module
$\mathcal {F}$
, we can define base extensions
$\mathcal {X}_{L}:=\mathcal {X} \hat {\otimes }_{K} L$
and
$\mathcal {F}_{L}$
of
$\mathcal {X}$
and
$\mathcal {F}$
functorially (see [Reference Bosch, Güntzer and Remmert10, Section 9.3.6] and [Reference Conrad17, §3.1]). If the extension
$L/K$
is finite, then they are just the fiber product and the pullback in the usual sense. Otherwise, it seems unclear if it has usual properties as a fiber product: for an open immersion
$j: \mathcal {U}\to \mathcal {X}$
, what we know in this case is that the base extension
$j_{L}:\mathcal {U}_{L} \to \mathcal {X}_{L}$
is also an open immersion if j is quasi-compact (for example, if
$\mathcal {X}$
is quasi-separated and
$\mathcal {U}$
is an admissible affinoid open subset) or a Zariski open immersion. At any rate, [Reference Bosch, Güntzer and Remmert10, Proposition 9.3.6/1 and Corollary 9.3.6/2] imply that the base extension takes any admissible affinoid covering of
$\mathcal {X}$
to that of
$\mathcal {X}_{L}$
. We write the set of L-valued points
$\mathcal {X}_{L}(L)$
also as
$\mathcal {X}(L)$
.
Let
$(R,M,\mathbf {T},\phi )$
be an input datum for the eigenvariety machine over K and let
$\mathcal {E}\to Z_{\phi }$
be the associated eigenvariety over
$X=\mathrm {Sp}(R)$
. We say that a K-algebra homomorphism
$\lambda : \mathbf {T}\to L$
is an L-valued eigensystem in M if there exist an element
$x\in X(L)$
given by a K-algebra homomorphism
$x^{*}: R\to L$
and a nonzero element m of
$M\hat {\otimes }_{R, x^{*}}L$
such that we have
$h m=\lambda (h) m$
for any
$h\in \mathbf {T}$
. It is said to be of finite slope if
$\lambda (\phi )\neq 0$
. Then there exists a natural bijection between
$\mathcal {E}(L)$
and the set of L-valued eigensystems
$ \lambda $
in M of finite slopes [Reference Buzzard12, Lemma 5.9]. We state the following lemma for reference, which is in fact shown in [Reference Buzzard12]:
Lemma 2.1. Let
$(R,M,\mathbf {T},\phi )$
be an input datum for the eigenvariety machine over K and let
$\mathcal {E}\to Z_{\phi }$
be the associated eigenvariety over
$X=\mathrm {Sp}(R)$
. Let
$L/K$
be an extension of complete valuation fields and take
$z\in \mathcal {E}(L)$
. Let
$x\in X(L)$
and
$y\in Z_{\phi }(L)$
be the images of z. Let
$\lambda : \mathbf {T}\to L$
be the L-valued eigensystem in M corresponding to z. Let m be a nonzero element of
$M\hat {\otimes }_{R,x^{*}} L$
satisfying
$h m=\lambda (h) m$
for any
$h\in \mathbf {T}$
. Take an admissible affinoid open subset V in the canonical admissible covering of
$Z_{\phi }$
satisfying
$y\in V(L)$
. Write
$f(V)=\mathrm {Sp}(A)$
. Suppose
$f(V)$
is connected. Let
$P(T)$
be the characteristic power series of
$\phi $
acting on M. Let
$Q(T)$
be the factor of
$P(T)$
in
$A\{\{T\}\}$
associated to V and let
$M_{A}=N\oplus F$
be the corresponding decomposition of
$M_{A}$
, as before. Then we have the following:
-
(1)
$\lambda (h)=h(z)$
in L, where
$h(z)$
is the specialization at z of the image of h by the map
$\mathbf {T}\to \mathcal {O}(\mathcal {E})$
. -
(2) The decomposition
is the one corresponding to the factor
$$ \begin{align*} M\hat{\otimes}_{R, x^{*}}L=N\otimes_{A, x^{*}}L\oplus F\hat{\otimes}_{A,x^{*}}L \end{align*} $$
$Q_{x}(T)$
of
$P_{x}(T)$
, where
$P_{x}(T)$
and
$Q_{x}(T)$
are the images of
$P(T)$
and
$Q(T)$
in
$L\{\{T\}\}$
by
$x^{*}$
, respectively.
-
(3)
$Q_{x}\left (\lambda (\phi )^{-1}\right )=0$
and
$m\in N{\otimes }_{A,x^{*}} L$
.
Proof. The first assertion follows from the proof of [Reference Buzzard12, Lemma 5.9]. The second one follows from [Reference Buzzard12, Lemma 2.13] and the uniqueness of the decomposition in [Reference Buzzard12, Theorem 3.3]. For the third one, note that the definition of the map
$\mathcal {E}(V)\to V$
implies
$Q_{x}\left (\lambda (\phi )^{-1}\right )=Q_{x}^{*}(\lambda (\phi ))=0$
. Since
$Q_{x}^{*}(\phi )m=Q_{x}^{*}(\lambda (\phi ))m=0$
, the second assertion implies
$m \in N{\otimes }_{A,x^{*}} L$
.
2.2 Lifting lemma à la Deligne and Serre
In this subsection, we consider the problem of converting a family of eigensystems into a family of eigenforms. First we show a local lemma in the following setting: Let L be a complete valuation field which is algebraically closed. Let
$(A,N,T,S,\varphi )$
be a tuple consisting of
-
• an L-affinoid algebra A,
-
• a projective A-module N of finite rank,
-
• a finite A-algebra T equipped with an A-algebra homomorphism
$T\to \mathrm {End}_{A}(N)$
, -
• an L-affinoid algebra S which is an integral domain, and
-
• a homomorphism
$\varphi : T\to S$
of L-affinoid algebras.
For any
$x\in \mathrm {Sp}(S)$
, we write
$m_{x}$
for the associated maximal ideal of S.
Lemma 2.2. Let L be a complete valuation field which is algebraically closed. Let
$(A,N,T,S,\varphi )$
be a tuple as before. Assume that for any
$x\in \mathrm {Sp}(S)$
, the induced map
$$ \begin{align*} \varphi(-)(x): T\to S/m_{x} \end{align*} $$
is an
$S/m_{x}$
-valued eigensystem in N. Namely, we assume that for any
$x\in \mathrm {Sp}(S)$
, there exists a nonzero element
$f_{x}\in N\otimes _{A} S/m_{x}$
satisfying
$(h\otimes 1) f_{x}=(1\otimes \varphi (h)(x))f_{x}$
for any
$h\in T$
.
-
(1) There exists a nonzero element
$F\in N\otimes _{A} S$
satisfying
$(h\otimes 1)F=(1\otimes \varphi (h))F$
for any
$h\in T$
. -
(2) Assume moreover that S is a principal ideal domain. We write
$F(x)$
for the image of F in
$N\otimes _{A} S/m_{x}$
. Then there exists F as in (1) satisfying
$F(x)\neq 0$
for any
$x\in \mathrm {Sp}(S)$
.
Proof. Put
$P=\mathrm {Ker}(\varphi : T\to S)$
, which is a prime ideal of T. Consider the multiplication map
$\mu :T\otimes _{A}T/P\to T/P$
, and put
$$ \begin{align*} Q=\mathrm{Ker}(\mu)=\mathrm{Ker}(T\otimes_{A} T/P\to T/P \to S). \end{align*} $$
Then the ideal Q is a minimal prime ideal. Indeed, since the A-algebra T is finite, the
$T/P$
-algebra
$T\otimes _{A} T/P$
is also finite, and thus the latter ring is a finite extension of a quotient of
$T/P$
. Since the quotient
$(T\otimes _{A} T/P)/Q$
is isomorphic to
$T/P$
, we have the inequality
$$ \begin{align*} \dim(T/P)\geq \dim (T\otimes_{A} T/P)\geq \mathrm{ht}(Q)+\dim (T/P), \end{align*} $$
which implies
$\mathrm {ht}(Q)=0$
.
The ideals
$n_{x}=\varphi ^{-1}(m_{x})$
and
$\bar {n}^{\prime }_{x}=(\varphi \circ \mu )^{-1}(m_{x})$
are maximal ideals of the rings T and
$T\otimes _{A} T/P$
, respectively. We write
$\bar {n}_{x}$
for the inverse image of
$m_{x}$
by the map
$T/P\to S$
, which is also a maximal ideal. Via the map
$1\otimes \varphi : T\otimes _{A} T/P\to T\otimes _{A} S$
, the ring
$T\otimes _{A} T/P$
acts on
$N\otimes _{A} S/m_{x}$
for any
$x\in \mathrm {Sp}(S)$
.
First we claim that
$\bar {n}^{\prime }_{x}=\mathrm {Ann}_{T\otimes _{A} T/P}(f_{x})$
. Since
$\bar {n}^{\prime }_{x}$
is a maximal ideal and
$f_{x}\neq 0$
, it is enough to show
$\bar {n}^{\prime }_{x}\subseteq \mathrm {Ann}_{T\otimes _{A} T/P}(f_{x})$
. Since L is algebraically closed, the ideal
$\mathrm {Im}(n_{x}\otimes _{A} T/P)+\mathrm {Im}(T\otimes _{A} \bar {n}_{x})$
is a maximal ideal contained in
$\bar {n}^{\prime }_{x}$
, and thus they are equal. For any
$h\in T$
, we denote its image in
$T/P$
by
$\bar {h}$
. Take elements
$h\in T$
and
$h^{\prime }\in n_{x}$
. We have
$\left (h\otimes \bar {h}^{\prime }\right ) f_{x}=0$
. On the other hand, we also have
$(h^{\prime }\otimes 1)f_{x}=(1\otimes \varphi (h^{\prime })(x))f_{x}=0$
by assumption. This implies
$\left (h^{\prime }\otimes \bar {h}\right )f_{x}=0$
, and the claim follows.
Next we claim that the localization
$(N\otimes _{A} T/P)_{Q}$
of the
$T\otimes _{A} T/P$
-module
$N\otimes _{A} T/P$
at Q is nonzero. Suppose the contrary. Since the
$T\otimes _{A} T/P$
-module
$N\otimes _{A} T/P$
is finite, we can find
$s\notin Q$
satisfying
$s(N\otimes _{A} T/P)=0$
. Take any
$x\in \mathrm {Sp}(S)$
. We have
$s(N\otimes _{A} T/n_{x})=0$
. Since L is algebraically closed, we have
$L=T/n_{x}=S/m_{x}$
, and we also see that
$s(N\otimes _{A} S/m_{x})=0$
. In particular, we have
$sf_{x}=0$
and
$s\in \mathrm {Ann}_{T\otimes _{A} T/P}(f_{x})=\bar {n}^{\prime }_{x}$
. Thus we obtain
$$ \begin{align*} s\in \bigcap_{x \in \mathrm{Sp}(S)} \bar{n}^{\prime}_{x}=\bigcap_{x \in \mathrm{Sp}(S)} (\varphi\circ \mu)^{-1}(m_{x})= (\varphi\circ \mu)^{-1}\left(\bigcap_{x \in \mathrm{Sp}(S)} m_{x}\right). \end{align*} $$
The assumption that S is a reduced L-affinoid algebra implies
$$ \begin{align*} \bigcap_{x \in \mathrm{Sp}(S)} m_{x}=0. \end{align*} $$
Hence
$s\in \mathrm {Ker}(\varphi \circ \mu )=Q$
, which is a contradiction.
Therefore we obtain
$Q\in \mathrm {Supp}_{T\otimes _{A} T/P}(N\otimes _{A} T/P)$
. Since Q is a minimal prime ideal, it is also contained in
$\mathrm {Ass}_{T\otimes _{A} T/P}(N\otimes _{A} T/P)$
. Namely, the prime ideal Q is written as
$Q=\mathrm {Ann}_{T\otimes _{A} T/P}(G)$
with some nonzero element G of
$N\otimes _{A} T/P$
. Since the A-module N is projective, the natural map
$1\otimes \varphi : N\otimes _{A} T/P\to N\otimes _{A} S$
is an injection. Thus the image
$F=(1\otimes \varphi )(G)$
is nonzero. Moreover, since
$h\otimes 1-1\otimes \bar {h}\in Q$
for any
$h\in T$
, we have the equality
$(h\otimes 1)G=\left (1\otimes \bar {h}\right )G$
. Hence we obtain
$(h\otimes 1)F=(1\otimes \varphi (h))F$
, and assertion (1) follows.
Now assume that S is a principal ideal domain. Then each maximal ideal
$m_{x}$
of S is generated by a single element
$t_{x}$
. Put
$$ \begin{align*} \Sigma(F)=\{x\in \mathrm{Sp}(S)\mid F(x)=0\}. \end{align*} $$
Since the A-module N is projective and the Krull dimension of S is no more than
$1$
, we see that
$\Sigma (F)$
is a finite set. For any
$x\in \Sigma (F)$
, the element F lies in
$\mathrm {Ker}(N\otimes _{A} S\to N\otimes _{A} S/m_{x})=m_{x}(N\otimes _{A} S)$
. By Krull’s intersection theorem, there exists a positive integer
$c_{x}$
satisfying
$F\in t_{x}^{c_{x}} (N\otimes _{A} S)\setminus t_{x}^{c_{x}+1} (N\otimes _{A} S)$
. Put
$F=t_{x}^{c_{x}} H$
with some nonzero element H of
$N\otimes _{A} S$
. We have
$H(x)\neq 0$
and
$\Sigma (H)\subsetneq \Sigma (F)$
. Since the S-module
$N\otimes _{A} S$
is torsion free, the element H also satisfies
$(h\otimes 1)H=(1\otimes \varphi (h))H$
for any
$h\in T$
. Repeating this, we can find F as in assertion (1) satisfying
$\Sigma (F)=\emptyset $
.
Remark 2.3. Let
$\mathrm {Sp}(S)$
be a connected affinoid subdomain of the unit disc
$\mathcal {D}_{\mathbb {C}_{p}}=\mathrm {Sp}\left (\mathbb {C}_{p}\langle T\rangle \right )$
. Note that
$\mathbb {C}_{p}\langle T\rangle $
is a principal ideal domain, since it is a unique factorization domain of Krull dimension
$1$
. [Reference Bosch, Güntzer and Remmert10, Proposition 7.2.2/1] implies that S is a regular ring of Krull dimension no more than
$1$
such that every maximal ideal is principal. Since
$\mathrm {Sp}(S)$
is connected, we see that S is a principal ideal domain. Hence the assumption of Lemma 2.2(2) is satisfied in this case.
We say that a rigid analytic variety X is principally refined if any admissible covering of X has a refinement by an admissible affinoid covering
$X=\bigcup _{i\in I} U_{i}$
such that the affinoid algebra of each affinoid open subset
$U_{i}$
in the refined covering is a principal ideal domain.
Remark 2.4. Remark 2.3 implies that any open subvariety of
$\mathcal {D}_{\mathbb {C}_{p}}$
is principally refined.
For the eigenvariety associated to an input datum
$(R,M,\mathbf {T},\phi )$
, Lemma 2.2 implies the following proposition:
Proposition 2.5. Let
$(R,M,\mathbf {T},\phi )$
be an input datum for the eigenvariety machine over K and let
$\mathcal {E}\to Z_{\phi }\to \mathrm {Sp}(R)$
be the associated eigenvariety. Let
$L/K$
be an extension of complete valuation fields such that L is algebraically closed. Let X be a smooth rigid analytic variety over L and let
$\varphi : X \to \mathcal {E}_{L}=\mathcal {E}\hat {\otimes }_{K} L$
be a morphism of rigid analytic varieties over L.
-
(1) There exist an admissible affinoid covering
$X=\bigcup _{i\in I} U_{i}$
and a nonzero element
$F_{i}\in M\hat {\otimes }_{R} \mathcal {O}(U_{i})$
for each
$i\in I$
satisfying
$(h\otimes 1)F_{i}=(1\otimes \varphi ^{*}(h))F_{i}$
for any
$h\in \mathbf {T}$
, where
$\varphi ^{*}: \mathbf {T}\to \mathcal {O}(\mathcal {E})\to \mathcal {O}(U_{i})$
is the map induced by
$\varphi $
. -
(2) Assume moreover that X is principally refined. We write
$k(x)$
for the residue field of
$x\in U_{i}$
and
$F_{i}(x)$
for the image of
$F_{i}$
in
$M\hat {\otimes }_{R} k(x)$
. Then we can find
$F_{i}$
as in (1) satisfying
$F_{i}(x)\neq 0$
for any
$x\in U_{i}$
.
Proof. Let
$\mathcal {C}$
be the canonical admissible covering of
$Z_{\phi }$
. For any
$V\in \mathcal {C}$
, we have the K-affinoid variety
$\mathcal {E}(V)=\mathrm {Sp}(\mathbf {T}(V))$
, as before. Then
$\mathcal {E}_{L} =\bigcup _{V\in \mathcal {C}} \mathcal {E}(V)_{L}$
is an admissible affinoid covering of
$\mathcal {E}_{L}$
. Let
$f:Z_{\phi }\to \mathrm {Sp}(R)$
be the natural projection and write
$f(V)=\mathrm {Sp}(A)$
. For any
$V\in \mathcal {C}$
such that
$f(V)$
is connected, take an admissible affinoid covering
$\varphi ^{-1}(\mathcal {E}(V)_{L})=\bigcup _{i\in I_{V}} U_{i}$
such that
$U_{i}=\mathrm {Sp}(S_{i})$
is connected for any
$i\in I_{V}$
. From the construction of the eigenvariety, we have a natural decomposition
$M\hat {\otimes }_{R} A=N\oplus F$
into closed A-submodules N and F. Note that the A-module N is finite and projective. Since the complete tensor product commutes with the direct sum, the
$S_{i}$
-module
$N\otimes _{A} S_{i}$
is a submodule of
$M\hat {\otimes }_{R} S_{i}$
.
For any
$i\in I_{V}$
, consider the natural map
$\varphi ^{*}: \mathbf {T}\to \mathbf {T}(V)\to S_{i}$
. We denote the map
$\mathbf {T}(V)\to S_{i}$
also by
$\varphi ^{*}$
. For any
$x\in U_{i}=\mathrm {Sp}(S_{i})$
, the composite
$\mathrm {Sp}(k(x))\to U_{i}\to \mathcal {E}_{L}$
corresponds to a
$k(x)$
-valued eigensystem of
$\mathbf {T}$
in M of finite slope. Namely, there exists a nonzero element
$g_{x}$
of
$M\hat {\otimes }_{R} k(x)=N{\otimes }_{A} k(x)\oplus F\hat {\otimes }_{A} k(x)$
satisfying
$(h\otimes 1)g_{x}=(1\otimes \varphi ^{*}(h)(x)) g_{x}$
for any
$h\in \mathbf {T}$
and
$(\phi \otimes 1)g_{x}\neq 0$
. Lemma 2.1(3) implies
$g_{x}\in N{\otimes }_{A} k(x)$
. Since
$U_{i}$
is connected and smooth, the ring
$S_{i}$
is an integral domain. Applying Lemma 2.2(1) to the tuple
$\left (A\hat {\otimes }_{K} L,N\hat {\otimes }_{K} L,\mathbf {T}(V)\hat {\otimes }_{K} L, S_{i},\varphi ^{*}\otimes 1\right )$
, we obtain a nonzero element
$G_{i}\in N\otimes _{A} S_{i}=\left (N\hat {\otimes }_{K} L\right )\hat {\otimes }_{A\hat {\otimes }_{K} L} S_{i}$
satisfying
$(h\otimes 1)G_{i}=(1\otimes \varphi ^{*}(h))G_{i}$
for any
$h\in \mathbf {T}$
. Setting
$F_{i}$
to be the image of
$G_{i}$
by the injection
$N\otimes _{A} S_{i} \to M\hat {\otimes }_{R} S_{i}$
assertion (1) follows.
For assertion (2), by assumption we may assume that each
$S_{i}$
is a principal domain. Then Lemma 2.2(2) allows us to find
$G_{i}$
satisfying in addition
$G_{i}(x)\neq 0$
for any
$x\in U_{i}$
. Since we have a commutative diagram
such that the horizontal arrows are injective, we obtain
$F_{i}(x)\neq 0$
for any
$x\in U_{i}$
.
2.3 Bellaïche and Chenevier’s argument
Let
$(R,M,\mathbf {T},\phi )$
be an input datum for the eigenvariety machine over K and let
$\mathcal {E}\to Z_{\phi }\to \mathrm {Sp}(R)$
be the associated eigenvariety. Let
$L/K$
be an extension of complete valuation fields. Put
$R_{L}=R\hat {\otimes }_{K} L$
. Let X be a rigid analytic variety over L equipped with a morphism
$\kappa : X\to \mathrm {Sp}(R_{L})$
. For any
$x\in X$
, we have a natural ring homomorphism
$\kappa ^{*}(x): R\to k(x)$
. A ring homomorphism
$\varphi : \mathbf {T}\to \mathcal {O}(X)$
is said to be a family of eigensystems in M over X if, for any
$x\in X$
, there exists a nonzero element
$f_{x}$
of
$M\hat {\otimes }_{R,\kappa ^{*}(x)} k(x)$
such that
$(h\otimes 1)f_{x}=(1\otimes \varphi (h)(x))f_{x}$
for any
$h\in \mathbf {T}$
. It is said to be of finite slopes if
$\varphi (\phi )(x)\neq 0$
for any
$x\in X$
. This is the same as saying that
$\varphi (\phi )\in \mathcal {O}(X)^{\times }$
. In this subsection, we show that we can convert a family of eigensystems of finite slopes over a reduced base space into a morphism to the eigenvariety, following [Reference Bellaïche and Chenevier7, Proposition 7.2.8]. First we recall the following lemma:
Lemma 2.6.
-
(1) Let
$f:X\to Y$
be a morphism of rigid analytic varieties over L with X reduced. Let Z be a closed analytic subvariety of Y. Suppose
$f(X)\subseteq Z$
. Then f factors through Z. -
(2) Let
$f,f^{\prime }:X\to Y$
be two morphisms of rigid analytic varieties over L with X reduced and Y separated. Suppose that these morphisms define the same map between the underlying sets. Then
$f=f^{\prime }$
.
Proof. For the first assertion, we may assume that
$X=\mathrm {Sp}(R_{1})$
,
$Y=\mathrm {Sp}(R_{2})$
, and
$Z=\mathrm {Sp}(R_{2}/I)$
for some ideal I of
$R_{2}$
. Consider the associated ring homomorphism
$f^{*}: R_{2}\to R_{1}$
and put
$J=\mathrm {Ker}(f^{*})$
. By assumption, every maximal ideal m of
$R_{1}$
satisfies
$(f^{*})^{-1}(m)\supseteq I$
. Since
$R_{1}$
is Jacobson and reduced, we obtain
$$ \begin{align*} I\subseteq \bigcap_{m\in \mathrm{Sp}\left(R_{1}\right)} (f^{*})^{-1}(m)=(f^{*})^{-1}\left(\bigcap_{m\in \mathrm{Sp}\left(R_{1}\right)} m\right)=(f^{*})^{-1}(0)=J. \end{align*} $$
Hence assertion (1) follows. Assertion (2) follows from the first one applied to
$(f,f^{\prime }):X\to Y\times _{L} Y$
and the diagonal
$Y\to Y\times _{L} Y$
.
Proposition 2.7. Let
$(R,M,\mathbf {T},\phi )$
be an input datum for the eigenvariety machine over K and let
$\mathcal {E}\to Z_{\phi }\to \mathrm {Sp}(R)$
be the associated eigenvariety. Let
$L/K$
be an extension of complete valuation fields. Let X be a reduced rigid analytic variety over L equipped with a morphism
$\kappa : X\to \mathrm {Sp}(R_{L})$
. Suppose that we have a family of eigensystems of finite slopes
$\varphi : \mathbf {T}\to \mathcal {O}(X)$
in M over X. Then there exists a unique morphism
$\Phi :X\to \mathcal {E}_{L}$
such that the diagram
is commutative and, for any
$x\in X$
, the eigensystem over
$k(x)$
corresponding to
$\mathrm {Sp}(k(x))\to X\overset {\Phi }{\to } \mathcal {E}_{L}$
is the map
$\varphi (-)(x): \mathbf {T}\to k(x)$
.
Proof. Let
$\mathcal {C}$
be the canonical admissible covering of
$Z_{\phi }$
. Take any
$V=\mathrm {Sp}(B)\in \mathcal {C}$
and put
$f(V)=\mathrm {Sp}(A)$
as in the proof of Proposition 2.5. Let I be a finite subset of
$\mathbf {T}$
such that its image in
$\mathbf {T}(V)$
is a system of generators of the finite B-algebra
$\mathbf {T}(V)$
. We denote by
$\mathbb {A}^{I}_{V_{L}}$
the affine space over
$V_{L}=V\hat {\otimes }_{K} L$
whose variables are indexed by I. We have a morphism of rigid analytic varieties
$$ \begin{align*} i_{V,I}: \mathcal{E}(V)_{L}\to \mathbb{A}^{I}_{V_{L}},\quad z\mapsto (h(z))_{h\in I}. \end{align*} $$
From the definition of I, we see that the map
$i_{V,I}$
is a closed immersion.
On the other hand, we also have a morphism of rigid analytic varieties
$$ \begin{align*} \mu: X\to \mathrm{Sp}(R_{L})\times \mathbb{A}_{L}^{1}, \quad x\mapsto \left(\kappa(x),\varphi(\phi)^{-1}(x)\right). \end{align*} $$
Let
$P(T)\in R\{\{T\}\}$
be the characteristic power series of
$\phi $
acting on M. For any
$x\in X$
, let
$P_{x}(T)$
be the image of
$P(T)$
in
$k(x)\{\{T\}\}$
by the map
$\kappa ^{*}(x): R\to k(x)$
. By [Reference Buzzard12, Lemma 2.13], it is the characteristic power series of
$\phi $
acting on
$M\hat {\otimes }_{R,\kappa ^{*}(x)} k(x)$
. By assumption, there exists a nonzero element
$g_{x}$
of
$M\hat {\otimes }_{R,\kappa ^{*}(x)} k(x)$
satisfying
$$ \begin{align*} (h\otimes 1) g_{x}=(1\otimes \varphi(h)(x))g_{x}. \end{align*} $$
Then Lemma 2.1(3) implies
$P_{x}\left (\varphi (\phi )(x)^{-1}\right )=0$
. Using the assumption that X is reduced and Lemma 2.6(1), we see that the morphism
$\mu $
factors through
$Z_{\phi ,L}$
.
For any
$V\in \mathcal {C}$
, put
$X_{V_{L}}=\mu ^{-1}(V_{L})$
. For any I as before, we consider the morphism of rigid analytic varieties over
$V_{L}$
$$ \begin{align*} j_{V,I}: X_{V_{L}} \to \mathbb{A}^{I}_{V_{L}},\quad x\mapsto (\varphi(h)(x))_{h\in I}. \end{align*} $$
By [Reference Buzzard12, Lemma 5.9] and Lemma 2.1(1), for any
$x\in X_{V_{L}}$
there exists a unique point
$z_{x}\in \mathcal {E}(k(x))$
satisfying
$\varphi (h)(x)=h(z_{x})$
for any
$h\in \mathbf {T}$
. We claim that
$z_{x}\in \mathcal {E}(V)_{L}$
. Indeed, we may assume that
$f(V)$
is connected. Let
$Q(T)$
be the factor of
$P(T)$
corresponding to V and let
$Q_{x}(T)$
be its image by
$\kappa ^{*}(x)$
. Let N be the direct summand of
$M_{A}$
corresponding to V. For any
$x\in X_{V_{L}}$
, we have
$\mu (x)\in V_{L}$
and
$Q_{x}\left (\varphi \left (\phi ^{-1}\right ) (x)\right )=Q^{*}_{x}(\varphi (\phi )(x))=0$
. Hence
$Q^{*}_{x}(\phi )g_{x}=0$
and thus
$g_{x}\in N\otimes _{A} k(x)$
. From the proof of [Reference Buzzard12, Lemma 5.9], this implies
$z_{x}\in \mathcal {E}(V)_{L}$
, and the claim follows.
In particular, we have
$j_{V,I}(x)=i_{V,I}(z_{x})$
for any
$x\in X_{V_{L}}$
and thus
$j_{V,I}\left (X_{V_{L}}\right )\subseteq i_{V,I}(\mathcal {E}(V)_{L})$
. Since
$i_{V,I}$
is a closed immersion and
$X_{V_{L}}$
is reduced, Lemma 2.6(1) yields a unique morphism
$\Phi _{V,I}: X_{V_{L}}\to \mathcal {E}(V)_{L}$
over
$V_{L}$
which makes the following diagram commutative:
We claim that the morphism
$\Phi _{V,I}$
is independent of the choice of a finite subset I of
$\mathbf {T}$
as before. Indeed, for any
$x\in X_{V_{L}}$
, we have
$\Phi _{V,I}(x)=i_{V,I}^{-1}\left (j_{V,I}(x)\right )=z_{x}$
, which depends only on x. Since X is reduced and
$\mathcal {E}$
is separated, Lemma 2.6(2) implies the claim. Moreover, for the same reason we can glue the morphisms
$\Phi _{V,I}$
along
$V\in \mathcal {C}$
and obtain a morphism
$\Phi : X\to \mathcal {E}_{L}$
. Since the requirement on
$\Phi $
in the proposition is the same as
$\Phi (x)=z_{x}$
, it is satisfied by the morphism
$\Phi $
we have constructed. Lemma 2.6(2) ensures uniqueness.
3 Hilbert eigenvariety
3.1 Hilbert modular varieties
Let p be a rational prime. Let F be a totally real number field of degree g which is unramified over p. We denote its ring of integers by
$\mathfrak {o}=\mathcal {O}_{F}$
and its different by
$\mathcal {D}_{F}$
. For any integer N, we put
$$ \begin{align*} U_{N}=\left\{\epsilon\in \mathcal{O}_{F}^{\times}\mid \epsilon\equiv 1 \bmod N\right\}. \end{align*} $$
For any prime ideal
$\mathfrak {p} \mid p$
of
$\mathcal {O}_{F}$
, let
$f_{\mathfrak {p}}$
be the residue degree of
$\mathfrak {p}$
.
Fix a finite extension
$K/\mathbb {Q}_{p}$
in
$\bar {\mathbb {Q}}_{p}$
such that
$F\otimes K$
splits completely. We denote by
$\mathcal {O}_{K}$
the ring of integers of K, by
$m_{K}$
the maximal ideal of
$\mathcal {O}_{K}$
, by
$\pi _{K}$
a uniformizer of K, by e the absolute ramification index of K, by k the residue field of K, and by
$W=W(k)$
the Witt ring of k. Let
$v_{p}$
be the additive valuation on K normalized as
$v_{p}(p)=1$
. For any nonnegative real number i, we put
$$ \begin{align*} m_{K}^{\geqslant i}=\left\{x\in \mathcal{O}_{K}\mid v_{p}(x)\geq i\right\}, \qquad \mathcal{O}_{K,i}=\mathcal{O}_{K}/m_{K}^{\geqslant i},\qquad \mathscr{S}_{i}=\mathrm{Spec}\left(\mathcal{O}_{K,i}\right). \end{align*} $$
For any extension
$L/K$
of valuation fields, we consider the valuation on L extending
$v_{p}$
and define
$\mathcal {O}_{L}$
,
$m_{L}$
,
$m_{L}^{\geqslant i}$
,
$\mathcal {O}_{L,i}$
, and
$\mathscr {S}_{L,i}=\mathrm {Spec}(\mathcal {O}_{L,i})$
similarly. For any element
$x\in \mathcal {O}_{L,1}$
, we define the truncated valuation
$v_{p}(x)$
by
$$ \begin{align*} v_{p}(x)=\min\left\{v_{p}\left(\hat{x}\right),1\right\} \end{align*} $$
with any lift
$\hat {x}\in \mathcal {O}_{L}$
of x. For any
$x\in L$
, we define the absolute value of x by
$\lvert x\rvert =p^{-v_{p}(x)}$
.
We denote by
$\mathbb {B}_{F}$
the set of embeddings
$F\to K$
and by
$\mathbb {B}_{\mathfrak {p}}$
the subset of
$\mathbb {B}_{F}$
consisting of embeddings which factor through the completion
$F_{\mathfrak {p}}$
. The set
$\mathbb {B}_{F}$
is decomposed as
$$ \begin{align*} \mathbb{B}_{F}=\coprod_{\mathfrak{p}\mid p} \mathbb{B}_{\mathfrak{p}}. \end{align*} $$
For any subset X of F, we denote by
$X^{+}$
the subset of totally positive elements of X. Put
$F_{\mathbb {R}}=F\otimes \mathbb {R}$
and
$F^{*}_{\mathbb {R}}=\mathrm {Hom}_{\mathbb {Q}}(F,\mathbb {R})$
. We denote by
$F^{*,+}_{\mathbb {R}}$
the subset of
$F^{*}_{\mathbb {R}}$
consisting of linear forms which maps the subset
$F^{\times ,+}$
to
$\mathbb {R}_{>0}$
. The group
$U_{N}$
acts on F and
$F^{*,+}_{\mathbb {R}}$
through
$\epsilon \mapsto \epsilon ^{2}$
.
Let
$\mathfrak {c}$
be any nonzero fractional ideal of F. For any fractional ideals
$\mathfrak {a}, \mathfrak {b}$
of F satisfying
$\mathfrak {a}\mathfrak {b}^{-1}=\mathfrak {c}$
, we denote by
$\mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
the set of rational polyhedral cone decompositions
$\mathscr {C}=\{\sigma \}_{\sigma \in \mathscr {C}}$
of
$F^{*,+}_{\mathbb {R}}$
which is projective and smooth with respect to the lattice
$\mathrm {Hom}(\mathfrak {a}\mathfrak {b},\mathbb {Z})$
such that the elements of
$\mathscr {C}$
are permuted by the action of
$U_{N}$
, the set
$\mathscr {C}/U_{N}$
is finite, and for any
$\epsilon \in U_{N}$
and
$\sigma \in \mathscr {C}$
,
$\epsilon (\sigma )\cap \sigma \neq \emptyset $
implies
$\epsilon =1$
, as in [Reference Hida31, §4.1.4]. Here we adopt the convention that
$\sigma $
is an open cone. Note that any two elements of
$\mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
have a common refinement which belongs to
$\mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
. For any such pair
$(\mathfrak {a},\mathfrak {b})$
, we fix once and for all a rational polyhedral cone decomposition
$\mathscr {C}(\mathfrak {a},\mathfrak {b})\in \mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
and put
$\mathscr {D}(\mathfrak {c})=\left \{\mathscr {C}(\mathfrak {a},\mathfrak {b})\mid \mathfrak {a}\mathfrak {b}^{-1}=\mathfrak {c}\right \}$
.
3.1.1 Hilbert–Blumenthal abelian varieties
Let
$N\geq 4$
be an integer with
$p\nmid N$
and let
$\mathfrak {c}$
be a nonzero fractional ideal of F. Let S be a scheme over
$\mathcal {O}_{K}$
. A Hilbert–Blumenthal abelian variety over S, which we abbreviate as ‘HBAV’, is a quadruple
$(A,\iota ,\lambda ,\psi )$
such that the following are true:
-
• A is an abelian scheme over S of relative dimension g.
-
•
$\iota : \mathcal {O}_{F}\to \mathrm {End}_{S}(A)$
is a ring homomorphism. -
•
$\lambda $
is a
$\mathfrak {c}$
-polarization. Namely,
$\lambda :A\otimes _{\mathcal {O}_{F}} \mathfrak {c}\simeq A^{\vee }$
is an isomorphism of abelian schemes to the dual abelian scheme
$A^{\vee }$
compatible with
$\mathcal {O}_{F}$
-action such that the map induces an isomorphism of
$$ \begin{align*} \mathrm{Hom}_{\mathcal{O}_{F}}\left(A,A^{\vee}\right)\simeq \mathrm{Hom}_{\mathcal{O}_{F}}\left(A,A\otimes_{\mathcal{O}_{F}} \mathfrak{c}\right),\quad f\mapsto \lambda^{-1}\circ f, \end{align*} $$
$\mathcal {O}_{F}$
-modules with notion of positivity
$\left (\mathcal {P}_{A},\mathcal {P}_{A}^{+}\right )\simeq \left (\mathfrak {c},\mathfrak {c}^{+}\right )$
. Here
$\mathcal {P}_{A}$
denotes the
$\mathcal {O}_{F}$
-module of symmetric
$\mathcal {O}_{F}$
-homomorphisms from A to
$A^{\vee }$
,
$\mathcal {P}_{A}^{+}$
is the subset of
$\mathcal {O}_{F}$
-linear polarizations, and any element
$\gamma \in \mathfrak {c}$
is identified with the element
$(x\mapsto x\otimes \gamma )$
of
$\mathrm {Hom}_{\mathcal {O}_{F}}\left (A,A\otimes _{\mathcal {O}_{F}} \mathfrak {c}\right )$
.
-
•
$\psi : \mathcal {D}_{F}^{-1}\otimes \mu _{N}\to A$
is an
$\mathcal {O}_{F}$
-linear closed immersion of group schemes, which we call a
$\Gamma _{00}(N)$
-structure.
Note that for such data, the
$\mathcal {O}_{F}\otimes \mathcal {O}_{S}$
-module
$\mathrm {Lie}(A)$
is locally free of rank
$1$
[Reference Deligne and Pappas20, Corollaire 2.9]. While the notion of an HBAV depends on
$\mathfrak {c}$
, we will omit the reference to
$\mathfrak {c}$
when there is no risk of confusion.
Let
$L/K$
be an extension of complete valuation fields and let
$\mathfrak {p}$
be a prime ideal of
$\mathcal {O}_{F}$
dividing p. Let
$\mathcal {G}$
be a finite flat group scheme over
$ \mathcal {O}_{L}$
with an
$\mathcal {O}_{F_{\mathfrak {p}}}$
-action. We have decompositions
$$ \begin{align*} \omega_{\mathcal{G}}=\bigoplus_{\beta\in \mathbb{B}_{\mathfrak{p}}} \omega_{\mathcal{G},\beta},\qquad \mathrm{Lie}\left(\mathcal{G}\times \mathscr{S}_{L,n}\right)=\bigoplus_{\beta\in \mathbb{B}_{\mathfrak{p}}} \mathrm{Lie}\left(\mathcal{G}\times \mathscr{S}_{L,n}\right)_{\beta} \end{align*} $$
according with the decomposition
$\mathcal {O}_{F_{\mathfrak {p}}}\otimes W\simeq \prod _{\beta \in \mathbb {B}_{\mathfrak {p}}} W$
. Write
$\omega _{\mathcal {G},\beta }\simeq \bigoplus _{i} \mathcal {O}_{L}/(a_{i})$
with some
$a_{i}\in \mathcal {O}_{L}$
, and define the
$\beta $
-degree of
$\mathcal {G}$
by
$\deg _{\beta }(\mathcal {G})=\sum _{i} v_{p}(a_{i})$
. Similarly, for any finite flat group scheme
$\mathcal {H}$
over
$\mathcal {O}_{L}$
with an
$\mathcal {O}_{F}$
-action, we have decompositions
$$ \begin{align*} \mathcal{H}=\bigoplus_{\mathfrak{p}\mid p} \mathcal{H}_{\mathfrak{p}},\qquad \omega_{\mathcal{H}}=\bigoplus_{\beta\in \mathbb{B}_{F}} \omega_{\mathcal{H},\beta} \end{align*} $$
such that
$\mathcal {H}_{\mathfrak {p}}$
is a finite flat closed subgroup scheme of
$\mathcal {H}$
over
$\mathcal {O}_{L}$
and
$\omega _{\mathcal {H},\beta }=\omega _{\mathcal {H}_{\mathfrak {p}},\beta }$
for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
. We put
$\deg _{\beta }(\mathcal {H})=\deg _{\beta }\left (\mathcal {H}_{\mathfrak {p}}\right )$
for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
.
Suppose that
$\mathcal {G}$
is a truncated Barsotti–Tate group of level n, height h, and dimension d over
$\mathcal {O}_{L}$
. For the p-torsion part
$\mathcal {G}[p]$
of
$\mathcal {G}$
, the Lie algebra
$\mathrm {Lie}\left (\mathcal {G}^{\vee }[p] \times \mathscr {S}_{L,1}\right )$
is a free
$\mathcal {O}_{L,1}$
-module of rank
$h-d$
. The Verschiebung of
$\mathcal {G}^{\vee }[p]\times \mathscr {S}_{L,1}$
yields a map
$$ \begin{align*} \mathrm{Lie}\left(V_{\mathcal{G}^{\vee}[p]\times \mathscr{S}_{L,1}}\right): \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times \mathscr{S}_{L,1}\right)^{(p)} \to \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times \mathscr{S}_{L,1}\right). \end{align*} $$
Then the Hodge height
$\mathrm {Hdg}(\mathcal {G})$
of
$\mathcal {G}$
is by definition the truncated valuation for
$v_{p}$
of the determinant of a representing matrix of this map. Moreover, if the ring
$\mathcal {O}_{F_{\mathfrak {p}}}$
acts on
$\mathcal {G}$
, then this map is also decomposed as the direct sum of maps
$$ \begin{align*} \mathrm{Lie}\left(V_{\mathcal{G}^{\vee}[p]\times\mathscr{S}_{L,1}}\right)_{\beta}: \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times\mathscr{S}_{L,1}\right)_{\sigma^{-1}\circ\beta}\to \mathrm{Lie}\left(\mathcal{G}^{\vee}[p]\times\mathscr{S}_{L,1}\right)_{\beta}, \end{align*} $$
where
$\sigma $
denotes the natural lift to W of the pth-power Frobenius map on k. If the
$\mathcal {O}_{L,1}$
-module
$\mathrm {Lie}\left (\mathcal {G}^{\vee }[p]\times \mathscr {S}_{L,1}\right )_{\beta }$
is free for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
, we define the
$\beta $
-Hodge height
$\mathrm {Hdg}_{\beta }(\mathcal {G})$
of
$\mathcal {G}$
as the truncated valuation of the determinant of the map
$ \mathrm {Lie}\left (V_{\mathcal {G}^{\vee }[p]\times \mathscr {S}_{L,1}}\right )_{\beta }$
. These assumptions are satisfied when
$\mathcal {G}$
is the
$\mathfrak {p}^{n}$
-torsion part
$A[\mathfrak {p}^{n}]=A[p^{n}]_{\mathfrak {p}}$
of an HBAV A over
$\mathcal {O}_{L}$
or, more generally, when
$\mathcal {G}$
is an
$\mathcal {O}_{F_{\mathfrak {p}}}$
-
$\mathrm {ADBT}_{n}$
[Reference Hattori29, §3]. For any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
, we put
$ \mathrm {Hdg}_{\beta }(A)=\mathrm {Hdg}_{\beta }(A[\mathfrak {p}^{n}])$
.
3.1.2 Moduli spaces and toroidal compactifications
Let
$M(\mu _{N},\mathfrak {c})$
be the Hilbert modular variety over
$\mathcal {O}_{K}$
which parametrizes the isomorphism classes of HBAVs
$(A,\iota , \lambda , \psi )$
such that
$ \lambda $
is a
$\mathfrak {c}$
-polarization and
$\psi $
is a
$\Gamma _{00}(N)$
-structure. The scheme
$M(\mu _{N},\mathfrak {c})$
is smooth over
$\mathcal {O}_{K}$
[Reference Goren26, Chapter 3, Theorem 6.9]. We denote by
$A^{\mathrm {un}}$
the universal HBAV over
$M(\mu _{N},\mathfrak {c})$
.
An unramified cusp for
$M(\mu _{N},\mathfrak {c})$
is a triple
$(\mathfrak {a},\mathfrak {b}, \phi _{N})$
of fractional ideals
$\mathfrak {a},\mathfrak {b}$
of F satisfying
$\mathfrak {a}\mathfrak {b}^{-1}=\mathfrak {c}$
and an isomorphism of
$\mathcal {O}_{F}$
-modules
$$ \begin{align*} \phi_{N}: \mathfrak{a}^{-1}/N\mathfrak{a}^{-1}\simeq \mathcal{O}_{F}/N\mathcal{O}_{F}. \end{align*} $$
For each cusp, we have a Tate object
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
over a certain base scheme [Reference Rapoport36, §4], which is used to construct a toroidal compactification
$\bar {M}(\mu _{N},\mathfrak {c})$
of
$M(\mu _{N},\mathfrak {c})$
. We recall the definition for unramified cusps. Put
$M=\mathfrak {a} \mathfrak {b}$
,
$M_{\mathbb {R}}=M\otimes \mathbb {R}$
, and
$M^{*}_{\mathbb {R}}=\mathrm {Hom}(M,\mathbb {R})$
. We identify
$M\otimes \mathbb {Q}$
with F. Then any
$\mathscr {C}\in \mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
gives a rational polyhedral cone decomposition of
$$ \begin{align*} M^{*,+}_{\mathbb{R}}=\left\{f\in M^{*}_{\mathbb{R}}\mid f(M^{+})\subseteq \mathbb{R}_{>0}\right\}. \end{align*} $$
For each
$\sigma \in \mathscr {C}$
, put
$$ \begin{align*} \sigma^{\vee} = \{ m\in M_{\mathbb{R}}\mid l(m)\geq 0 \text{ for any } l\in \sigma\}. \end{align*} $$
Then we have an affine torus embedding
$$ \begin{align*} S=\mathrm{Spec}(\mathcal{O}_{K}[q^{m}\mid m\in M])\to S_{\sigma}= \mathrm{Spec}\left(\mathcal{O}_{K}\left[q^{m}\mid m \in M\cap \sigma^{\vee}\right]\right). \end{align*} $$
The affine schemes
$\{S_{\sigma }\}_{\sigma \in \mathscr {C}}$
can be glued via
$S_{\sigma }\cap S_{\tau }=S_{\sigma \cap \tau }$
to define a torus embedding
$S \to S_{\mathscr {C}}$
. We denote by
$S^{\infty }_{\sigma }$
and
$S^{\infty }_{\mathscr {C}}=\bigcup _{\sigma \in \mathscr {C}} S^{\infty }_{\sigma }$
the complements of S in these embeddings with reduced structures. The formal completions along these closed subschemes are denoted by
$\hat {S}_{\sigma }=\mathrm {Spf}\left (\hat {R}_{\sigma }\right )$
and
$\hat {S}_{\mathscr {C}}$
. By assumption, we can construct the quotient
$\hat {S}_{\mathscr {C}}/U_{N}$
by regluing
$\left \{\hat {S}_{\sigma }\right \}_{\sigma \in \mathscr {C}}$
via the action
$\epsilon : \hat {S}_{\sigma }\simeq \hat {S}_{\epsilon \sigma }$
for any
$\epsilon \in U_{N}$
. The closed subscheme
$S^{\infty }_{\sigma }$
is defined by a principal ideal
$\hat {I}_{\sigma }$
of the ring
$\hat {R}_{\sigma }$
satisfying
$\sqrt {\hat {I}_{\sigma }}=\hat {I}_{\sigma }$
. The ring
$\hat {R}_{\sigma }$
is a Noetherian normal excellent ring which is complete with respect to the
$\hat {I}_{\sigma }$
-adic topology. Put
$\bar {S}_{\sigma }=\mathrm {Spec}\left (\hat {R}_{\sigma }\right )$
,
$\bar {S}_{\sigma }^{\infty }=V\left (\hat {I}_{\sigma }\right )$
, and
$\bar {S}^{0}_{\sigma }=\bar {S}_{\sigma }\setminus \bar {S}_{\sigma }^{\infty }$
, where the latter is an affine scheme and we denote its affine ring by
$\hat {R}_{\sigma }^{0}$
.
Note that the torus with character group
$\mathfrak {a}$
is
$(\mathfrak {a}\mathcal {D}_{F})^{-1}\otimes \mathbb {G}_{\mathrm {m}}$
. For any
$\eta \in \mathfrak {a}$
, we denote by
$\mathfrak {X}^{\eta }$
the element of
$\mathcal {O}\left ((\mathfrak {a}\mathcal {D}_{F})^{-1}\otimes \mathbb {G}_{\mathrm {m}}\right )$
which the character
$\eta $
defines. We have an
$\mathcal {O}_{F}$
-linear homomorphism
$$ \begin{align*} q: \mathfrak{b} \to (\mathfrak{a}\mathcal{D}_{F})^{-1}\otimes \mathbb{G}_{\mathrm{m}}\left(\bar{S}_{\sigma}^{0}\right) \end{align*} $$
defined by
$\xi \mapsto \left (\mathfrak {X}^{\eta }\mapsto q^{\xi \eta }\right )$
with
$\xi \in \mathfrak {b}$
and
$\eta \in \mathfrak {a}$
. By Mumford’s construction, we obtain the semi-abelian scheme
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
over
$\bar {S}_{\sigma }$
such that its restriction to
$\bar {S}_{\sigma }^{0}$
is an abelian scheme [Reference Rapoport36, §4]. It admits a natural
$\mathcal {O}_{F}$
-action. Over
$\bar {S}_{\sigma }^{0}$
, we have a natural exact sequence
which defines, for any unramified cusp
$(\mathfrak {a},\mathfrak {b},\phi _{N})$
, a
$\Gamma _{00}(N)$
-structure on
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)\rvert _{\bar {S}_{\sigma }^{0}}$
using
$\phi _{N}$
. Moreover, the natural isomorphism
$$ \begin{align*} \left((\mathfrak{a}\mathcal{D}_{F})^{-1}\otimes \mathbb{G}_{\mathrm{m}}\right)\otimes_{\mathcal{O}_{F}}\mathfrak{c} \to (\mathfrak{b}\mathcal{D}_{F})^{-1}\otimes \mathbb{G}_{\mathrm{m}} \end{align*} $$
induces a
$\mathfrak {c}$
-polarization
$$ \begin{align*} \lambda_{\mathfrak{a},\mathfrak{b}}: \mathrm{Tate}_{\mathfrak{a},\mathfrak{b}}(q)\rvert_{\bar{S}_{\sigma}^{0}} \otimes_{\mathcal{O}_{F}} \mathfrak{c}\to \mathrm{Tate}_{\mathfrak{b},\mathfrak{a}}(q)\rvert_{\bar{S}_{\sigma}^{0}} \simeq \left(\mathrm{Tate}_{\mathfrak{a},\mathfrak{b}}(q)\rvert_{\bar{S}_{\sigma}^{0}}\right)^{\vee}. \end{align*} $$
By these data we consider the Tate object
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)\rvert _{\bar {S}_{\sigma }^{0}}$
as an HBAV over
$\bar {S}_{\sigma }^{0}$
, which yields a morphism
$\bar {S} {}_{\sigma }^{0}\to M(\mu _{N},\mathfrak {c})$
. Then the toroidal compactification
$\bar {M}^{\mathscr {D}(\mathfrak {c})}(\mu _{N},\mathfrak {c})$
of
$M(\mu _{N},\mathfrak {c})$
over
$\mathcal {O}_{K}$
with respect to
$\mathscr {D}(\mathfrak {c})$
– which we also denote by
$\bar {M}(\mu _{N},\mathfrak {c})$
if no confusion will occur – is constructed in such a way as to satisfy the following (see [Reference Rapoport36, Théorème 6.18] and [Reference Dimitrov23, Théorème 7.2 (i)]):
-
•
$\bar {M}(\mu _{N},\mathfrak {c})$
is projective and smooth over
$\mathcal {O}_{K}$
. -
•
$M(\mu _{N},\mathfrak {c})$
is an open subscheme of
$\bar {M}(\mu _{N},\mathfrak {c})$
which is fiberwise dense and the complement D of
$M(\mu _{N},\mathfrak {c})$
is a normal crossing divisor. In particular,
$M(\mu _{N},\mathfrak {c})$
is quasi-compact. -
• The formal completion
$\bar {M}(\mu _{N},\mathfrak {c})\rvert ^{\wedge }_{D}$
of
$\bar {M}(\mu _{N},\mathfrak {c})$
along the boundary divisor D contains as a formal open subscheme, where the disjoint union runs over the set of isomorphism classes of unramified cusps.
$$ \begin{align*} \coprod \hat{S}_{\mathscr{C}(\mathfrak{a},\mathfrak{b})}/U_{N} \end{align*} $$
-
• The universal HBAV
$A^{\mathrm {un}}$
over
$M(\mu _{N},\mathfrak {c})$
extends to a semi-abelian scheme
$\bar {A}^{\mathrm {un}}$
with
$\mathcal {O}_{F}$
-action over
$\bar {M}(\mu _{N},\mathfrak {c})$
such that for any
$\sigma \in \mathscr {C}(\mathfrak {a},\mathfrak {b})$
, the pullback of
$\bar {A}^{\mathrm {un}}$
by the restriction to
$\bar {S}_{\sigma }^{0}$
of the unique algebraization
$\bar {S} {}_{\sigma }\to \bar {M}(\mu _{N},\mathfrak {c})$
of the map
$\hat {S}_{\sigma }\to \bar {M}(\mu _{N},\mathfrak {c})\rvert _{D}^{\wedge }$
for any unramified cusp
$(\mathfrak {a},\mathfrak {b},\phi _{N})$
is isomorphic to
$ \mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)\rvert _{\bar {S}_{\sigma }^{0}}$
.
3.1.3 Strict neighborhoods of the ordinary locus and their integral models
Let
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
be the p-adic formal completion of
$\bar {M}(\mu _{N},\mathfrak {c})$
. Let
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$
be its Raynaud generic fiber. Let
$ \mathcal {M}(\mu _{N},\mathfrak {c})$
be the analytification of the scheme
$M(\mu _{N},\mathfrak {c})\otimes _{\mathcal {O}_{K}}K$
, which is a Zariski open subvariety of
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$
. The semi-abelian scheme
$\bar {A}^{\mathrm {un}}$
defines semi-abelian objects
$\bar {\mathfrak {A}}^{\mathrm {un}}$
over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
and
$\bar {\mathcal {A}}^{\mathrm {un}}$
over
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$
by taking the p-adic completion and the Raynaud generic fiber. For the zero section e of
$\bar {A}^{\mathrm {un}}$
, put
$\omega _{\bar {A}^{\mathrm {un}}}=e^{*}\Omega ^{1}_{\bar {A}^{\mathrm {un}}/\bar {M}\left (\mu _{N},\mathfrak {c}\right )}$
. For any g-tuple
$\kappa =\left (k_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
in
$\mathbb {Z}$
, we define
$$ \begin{align*} \omega_{\bar{A}^{\mathrm{un}},\beta}=\omega_{\bar{A}^{\mathrm{un}}}\otimes_{\mathcal{O}_{F}, \beta}\mathcal{O}_{K},\qquad \omega^{\kappa}_{\bar{A}^{\mathrm{un}}}=\bigotimes_{\beta\in \mathbb{B}_{F}} \omega_{\bar{A} {}^{\mathrm{un}},\beta}^{\otimes k_{\beta}}. \end{align*} $$
We also define
$\omega _{\bar {\mathcal {A}}^{\mathrm {un}},\beta }$
and
$\omega ^{\kappa }_{\bar {\mathcal {A}}^{\mathrm {un}}}$
similarly. For any
$\beta \in \mathbb {B}_{F}$
, let
$h_{\beta }$
be the
$\beta $
-partial Hasse invariant, which is a section of the invertible sheaf
$\omega _{\bar {A}^{\mathrm {un}},\sigma ^{-1} \circ \beta }^{p}\otimes \omega _{\bar {A}^{\mathrm {un}},\beta }^{-1}$
on
$\bar {M}(\mu _{N},\mathfrak {c})\times \mathscr {S}_{1}$
[Reference Goren and Kassaei27, §2.5] (see also [Reference Andreatta and Goren2, §7] and [Reference Emerton, Reduzzi and Xiao24, §3.1]). For any extension
$L/K$
of complete valuation fields, any HBAV A over
$\mathcal {O}_{L}$
, and any
$\beta \in \mathbb {B}_{F}$
, consider the element P of
$\bar {\mathcal {M}} (\mu _{N},\mathfrak {c})(L)$
induced by A and a lift
$\tilde {h}_{\beta }$
of
$h_{\beta }$
as a section of
$\omega _{\bar {A}^{\mathrm {un}},\sigma ^{-1}\circ \beta }^{p}\otimes \omega _{\bar {A}^{\mathrm {un}},\beta }^{-1}$
over an open neighborhood of P. Then we have the equality of truncated valuations
$$ \begin{align*} \mathrm{Hdg}_{\beta}(A)=v_{p}\left(\tilde{h}_{\beta}(P)\right). \end{align*} $$
If
$P\in \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(L)$
corresponds to a semi-abelian scheme A over
$\mathcal {O}_{L}$
which is not an abelian scheme, then we put
$\mathrm {Hdg}_{\beta }(A)=v_{p}\left (\tilde {h}_{\beta }(P)\right )=0$
.
Let
$\underline {v}=\left (v_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
be a g-tuple in
$[0,1]\cap \mathbb {Q}$
. We denote by
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
and
$\mathcal {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
the admissible open subsets of
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$
and
$\mathcal {M}(\mu _{N},\mathfrak {c})$
defined by
$v_{p}\left (\tilde {h}_{\beta }(P)\right )\leq v_{\beta }$
for any
$\beta \in \mathbb {B}_{F}$
, respectively. Note that
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
is quasi-compact. We define its integral model
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
as follows: Write
$v_{\beta }=a_{\beta }/b_{\beta }$
with nonnegative integers
$a_{\beta }$
and
$b_{\beta }\neq 0$
. Take a formal open covering
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})=\bigcup \mathfrak {U}_{i}$
such that every
$h_{\beta }$
lifts to a section
$\tilde {h}_{\beta }$
on each
$\mathfrak {U}_{i}$
. Consider the formal scheme whose restriction to each
$\mathfrak {U}_{i}$
is the admissible blowup of
$\mathfrak {U}_{i}$
along the ideal
$\left (p^{a_{\beta }},\tilde {h}_{\beta }^{b_{\beta }}\right )$
, and its locus where this ideal is generated by
$\tilde {h}_{\beta }^{b_{\beta }}$
. Repeat this for any
$\beta \in \mathbb {B}_{F}$
and define
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
as the normalization in
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
of the resulting formal scheme. We denote the special fibers of
$\bar {M}(\mu _{N},\mathfrak {c})$
and
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c}) \left (\underline {v}\right )$
by
$\bar {M}(\mu _{N},\mathfrak {c})_{k}$
and
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )_{k}$
, respectively. We also denote by
$\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
the complement in
$\bar {\mathfrak {M}} (\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
of the boundary divisor
$D_{k}$
of the special fiber. Then the open immersion
$\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
is of finite type and [Reference de Jong19, Proposition 7.2.4] implies that
$\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )^{\mathrm {rig}}$
is quasi-compact.
Let v be an element of
$[0,1]\cap \mathbb {Q}$
. When
$v_{\beta }=v$
for any
$\beta \in \mathbb {B}_{F}$
, we write
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
as
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)$
. Moreover, we denote by
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
the quasi-compact admissible open subset defined similarly to
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)$
with the usual Hasse invariant
$$ \begin{align*} h_{\mathrm{tot}}=\prod_{\beta\in \mathbb{B}_{F}}h_{\beta} \end{align*} $$
instead of
$h_{\beta }$
s. We also define similar spaces for these two variants, such as
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v)$
and
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
. Note that
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
is just the formal open subscheme of
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
over which all the
$\beta $
-partial Hasse invariants are invertible.
The formal scheme
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
is also a formal open subscheme of
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
containing
$D_{k}$
. Let
$\mathrm {sp}:\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})_{k}$
be the specialization map with respect to
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
. Then [Reference de Jong19, Propositions 7.2.1 and 7.2.4] yields
$$ \begin{align} \mathfrak{M}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)^{\mathrm{rig}}=\bar{\mathcal{M}}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)\setminus \mathrm{sp}^{-1}(D_{k}). \end{align} $$
Let R be a topological
$\mathcal {O}_{K}$
-algebra which is idyllic with respect to the p-adic topology [Reference Abbes1, Définition 1.10.1]. By [Reference Abbes1, Corollaire 2.13.9], any morphism
$\hat {f}:\mathrm {Spf}(R)\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
has a unique algebraization
$f:\mathrm {Spec}(R)\to \bar {M}(\mu _{N},\mathfrak {c})$
, and we have a semi-abelian scheme
$G_{R}=f^{*}\bar {A}^{\mathrm {un}}$
over
$\mathrm {Spec}(R)$
. Taking the reduction modulo p, we see that
$\hat {f}$
factors through
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
if and only if
$G_{R}$
is ordinary.
Let
$\mathbf {NAdm}$
be the category of admissible p-adic formal
$\mathcal {O}_{K}$
-algebras R such that R is normal. Note that we have
$R[1/p]^{\circ }=R$
by [Reference Bosch, Güntzer and Remmert10, remark after Proposition 6.3.4/1]. By [Reference Rapoport36, Lemme 3.1], we can see as in [Reference Andreatta, Iovita and Pilloni3, Proposition 5.2.1.1] that any morphism
$\mathrm {Sp}(R[1/p])\to \mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )^{\mathrm {rig}}$
corresponds uniquely to an isomorphism class of an HBAV A over
$\mathrm {Spec}(R)$
such that
$\mathrm {Hdg}_{\beta }(A_{x})\leq v_{\beta }$
for
$x\in \mathrm {Sp}(R[1/p])$
.
We give a proof of the following lemma for lack of a reference:
Lemma 3.1. Let
$L/K$
be an extension of complete valuation fields. Let
$\mathcal {X}$
be a connected smooth rigid analytic variety over L and let
$\mathcal {F}$
be an invertible sheaf on
$\mathcal {X}$
. Suppose that
$f\in \mathcal {F}(\mathcal {X})$
vanishes on a nonempty admissible open subset
$\mathcal {U}$
of
$\mathcal {X}$
. Then
$f=0$
.
Proof. Take an admissible affinoid covering
$\mathcal {X}=\bigcup _{i\in I} \mathcal {X}_{i}$
such that
$\mathcal {X}_{i}$
is connected and
$\mathcal {F}$
is trivial on
$\mathcal {X}_{i}$
for any
$i\in I$
. We have
$\mathcal {X}_{i_{0}}\cap \mathcal {U}\neq \emptyset $
for some
$i_{0}$
. Then [Reference Fresnel and van der Put25, Exercise 4.6.3] implies
$f\rvert _{\mathcal {X}_{i_{0}}}=0$
.
Put
$I_{0}=\left \{i\in I\mid f\rvert _{\mathcal {X}_{i}}=0\right \}$
, which is nonempty. [Reference Fresnel and van der Put25, Exercise 4.6.3] also implies that
$\mathcal {X}_{i}\cap \mathcal {X}_{j}=\emptyset $
for any
$i\in I_{0}$
and
$j\in I_{1}:=I\setminus I_{0}$
. Then for the subsets
$$ \begin{align*} \mathcal{X}_{0}=\bigcup_{i\in I_{0}} \mathcal{X}_{i},\qquad \mathcal{X}_{1}=\bigcup_{i\in I_{1}} \mathcal{X}_{i} \end{align*} $$
and
$s\in \{0,1\}$
, the intersection
$\mathcal {X}_{s}\cap \mathcal {X}_{i}$
equals
$\mathcal {X}_{i}$
if
$i\in I_{s}$
and
$\emptyset $
if
$i\notin I_{s}$
. Hence
$\mathcal {X}=\mathcal {X}_{0}\coprod \mathcal {X}_{1}$
is an admissible covering of
$\mathcal {X}$
. Since
$\mathcal {X}$
is connected, we obtain
$\mathcal {X}=\mathcal {X}_{0}$
and
$f=0$
.
Lemma 3.2. Let
$L/K$
be an extension of complete valuation fields. Let
$\mathfrak {R}$
be an admissible formal
$\mathcal {O}_{L}$
-algebra such that
$\mathfrak {R}[1/p]$
is reduced. Suppose that
$\mathrm {Spec}(\mathfrak {R}/m_{L} \mathfrak {R})$
is reduced and connected. Then
$\mathrm {Sp}(\mathfrak {R}[1/p])$
is connected.
Proof. Take a surjection
$\mathcal {O}_{L}\langle T_{1},\dotsc ,T_{n}\rangle \to \mathfrak {R}$
. Then [Reference Bosch, Lütkebohmert and Raynaud11, Proposition 1.1] shows
$(\mathfrak {R}[1/p])^{\circ }=\mathfrak {R}$
. Moreover, by [Reference Bosch, Güntzer and Remmert10, remark after Proposition 6.3.4/1]
$\mathfrak {R}$
is integrally closed in
$\mathfrak {R}[1/p]$
. Thus we have bijections between the sets of connected components
$$ \begin{align*} \pi_{0}(\mathrm{Sp}(\mathfrak{R}[1/p]))\simeq \pi_{0}(\mathrm{Spec}(\mathfrak{R}))\simeq \pi_{0}(\mathrm{Spec}(\mathfrak{R}/m_{L} \mathfrak{R})), \end{align*} $$
from which the lemma follows.
Lemma 3.3. Let
$l/k$
be a finite extension and put
$W^{\prime }=W(l)$
. Let
$K^{\prime }$
be the composite field of K and
$\mathrm {Frac}(W^{\prime })$
in
$\bar {\mathbb {Q}}_{p}$
. Let R be a Noetherian W-algebra and let m be a maximal ideal of R with residue field l. Consider the natural maps
$$ \begin{align*} R\otimes_{W} W^{\prime}\to l,\qquad R\otimes_{W} \mathcal{O}_{K}\to l,\qquad R\otimes_{W} \mathcal{O}_{K^{\prime}}\to l \end{align*} $$
induced by
$R\to R/m=l$
. We denote the kernels of these maps by
$m^{\prime }$
, n, and
$n^{\prime }$
, respectively. Then the natural maps between complete local rings
$$ \begin{align*} R^{\wedge}_{m}\to (R\otimes_{W} W^{\prime})^{\wedge}_{m^{\prime}},\qquad R_{m}^{\wedge}\hat{\otimes}_{W}\mathcal{O}_{K}\to (R\otimes_{W}\mathcal{O}_{K})_{n}^{\wedge} \end{align*} $$
are bijective. In particular, we have isomorphisms
$$ \begin{align*} (R\otimes_{W} \mathcal{O}_{K})^{\wedge}_{n}\to \left(R\otimes_{W}\mathcal{O}_{K^{\prime}}\right)_{n^{\prime}}^{\wedge}\gets(R\otimes_{W} W^{\prime})_{m^{\prime}}^{\wedge}\hat{\otimes}_{W^{\prime}}\mathcal{O}_{K^{\prime}}. \end{align*} $$
Proof. For the first map, since R is Noetherian and
$R/m=l$
is perfect, there exists a unique section
$W^{\prime }\to R^{\wedge }_{m}$
of the natural surjection
$R^{\wedge }_{m}\to l$
. Thus we obtain a homomorphism of
$W^{\prime }$
-algebras
$R\otimes _{W} W^{\prime }\to R^{\wedge }_{m}$
. This induces a map
$(R\otimes _{W} W^{\prime })^{\wedge }_{m^{\prime }}\to R^{\wedge }_{m}$
, which gives the inverse of the first map.
For the second map, since
$p\in m$
and
$\mathcal {O}_{K}$
is totally ramified over W, we have
$n=(m,\pi _{K})$
. This yields
$R^{\wedge }_{m}\hat {\otimes }_{W}\mathcal {O}_{K}\simeq (R\otimes _{W} \mathcal {O}_{K})^{\wedge }_{n}$
. The last assertion follows by applying them to
$R\otimes _{W} \mathcal {O}_{K}$
and
$R\otimes _{W} W^{\prime }$
.
Lemma 3.4. Let
$\mathcal {X}_{v}$
be either of the rigid analytic varieties
$$ \begin{align*} \bar{\mathcal{M}}(\mu_{N},\mathfrak{c})(v),\qquad \bar{\mathcal{M}}(\mu_{N},\mathfrak{c})(v_{\mathrm{tot}}). \end{align*} $$
Then
$\mathcal {X}_{v}\hat {\otimes }_{K}{\mathbb {C}_{p}}$
is connected for any
$v\in [0,1)\cap \mathbb {Q}$
.
Proof. Since
$\mathcal {X}_{v}$
is separated, it is enough to show that for any sufficiently large finite extension
$K^{\prime }/K$
, the base extension
$\mathcal {X}_{v}\hat {\otimes }_{K}{K^{\prime }}$
is connected [Reference Conrad17, Theorem 3.2.1]. Replacing K by
$K^{\prime }$
, we may assume that
$K^{\prime }=K$
and every connected component of
$\mathcal {X}_{v}$
has a K-rational point.
By Ribet’s theorem (see [Reference Goren26, Chapter 3, Theorem 6.19]), the ordinary locus
${\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}$
is geometrically connected. Then
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}$
is also geometrically connected, since it is smooth over k and contains
${\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}$
as a dense open subset. Since
$\mathcal {X}_{0}$
is the tube for the immersion
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{k}\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
, [Reference Berthelot9, Proposition 1.3.3] yields the lemma for
$v=0$
.
Consider the case of
$v>0$
. Suppose that
$\mathcal {X}_{v}$
is not connected. Then we can take its connected component
$\mathcal {U}$
which does not intersect
$\mathcal {X}_{0}$
. Since
$\mathcal {U}$
is quasi-compact, there exists a finite admissible affinoid covering
$\mathcal {U}=\bigcup _{i=1}^{m} \mathcal {U}_{i}$
of
$\mathcal {U}$
such that any
$\beta $
-partial Hasse invariant can be lifted to a section over
$\mathcal {U}_{i}$
. Using the maximal modulus principle on each
$\mathcal {U}_{i}$
, we see that there exists a positive rational number
$\delta $
satisfying
$$ \begin{align*} \max\left\{\mathrm{Hdg}_{\beta}(x)\mid \beta\in \mathbb{B}_{F}\right\}\geq \delta \end{align*} $$
for any
$x\in \mathcal {U}$
. Then for any rational number
$\varepsilon $
satisfying
$0<\varepsilon <\delta $
, we have
$\mathcal {X}_{\varepsilon }\cap \mathcal {U}=\emptyset $
.
On the other hand, let us consider the specialization map
$$ \begin{align*} \mathrm{sp}:\bar{\mathcal{M}}(\mu_{N},\mathfrak{c})\to \bar{\mathfrak{M}}(\mu_{N},\mathfrak{c})_{k} \end{align*} $$
with respect to
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
. Take any
$P\in \mathcal {U}(K)$
and consider its specialization
$\bar {P}=\mathrm {sp}(P)$
. Since
$P\notin \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(0)$
, the point
$\bar {P}$
corresponds to an HBAV. By [Reference Goren and Kassaei27, (2.5.1)] and Lemma 3.3, the complete local ring of
$M(\mu _{N},\mathfrak {c})$
at
$\bar {P}$
is isomorphic to the ring
$\mathcal {O}_{K}\left [\left [t_{\beta }\mid \beta \in \mathbb {B}_{F}\right ]\right ]$
. Then [Reference de Jong19, Lemma 7.2.5] gives an identification
$$ \begin{align} \mathrm{sp}^{-1}\left(\bar{P}\right)=\prod_{\beta\in\mathbb{B}_{F}} \mathcal{A}_{\beta}[0,1), \end{align} $$
where for any interval I we denote by
$\mathcal {A}_{\beta } I$
the annulus over K with parameter
$t_{\beta }$
defined by the condition
$\left \lvert t_{\beta }\right \rvert \in I$
. By [Reference Goren and Kassaei27, §4.2], we may assume that the parameter
$t_{\beta }$
satisfies
$$ \begin{align} \mathrm{Hdg}_{\beta}(A)= \begin{cases} \min\{v_p(t_\beta(Q)),1\} & \left(\beta\in \tau\left(\bar{P}\right)\right) \\ 0 & \left(\beta\notin \tau\left(\bar{P}\right)\right) \end{cases} \end{align} $$
for any
$Q\in \mathrm {sp}^{-1}\left (\bar {P}\right )$
and for any
$\beta \in \mathbb {B}_{F}$
, where A is the HBAV corresponding to Q and
$\tau \left (\bar {P}\right )$
is defined by [Reference Goren and Kassaei27, (2.3.3)].
Suppose
$\mathcal {X}_{v}=\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)$
. For any positive rational number
$\varepsilon $
, put
$$ \begin{align*} \mathrm{sp}^{-1}\left(\bar{P}\right)(\varepsilon)^{\prime}=\prod_{\beta\in\tau\left(\bar{P}\right)} \mathcal{A}_{\beta}\left[p^{-\varepsilon},1\right)\times \prod_{\beta\notin\tau\left(\bar{P}\right)} \mathcal{A}_{\beta}[0,1). \end{align*} $$
Since
$\mathrm {sp}^{-1}\left (\bar {P}\right )(v)^{\prime }$
is a connected admissible open subset of
$\mathcal {X}_{v}$
containing P, it is contained in
$\mathcal {U}$
. However, for any
$\varepsilon $
satisfying
$\varepsilon <\min \{\delta ,v\}$
, we have
$$ \begin{align*} \emptyset \neq \mathrm{sp}^{-1}\left(\bar{P}\right)(\varepsilon)^{\prime}\subseteq \mathcal{X}_{\varepsilon}\cap \mathcal{U}, \end{align*} $$
which is a contradiction.
When
$\mathcal {X}_{v}=\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
, take any rational number
$\varepsilon $
satisfying
$0<\varepsilon <\min \{\delta ,v\}$
and
$v_{p}\left (t_{\beta }(P)\right )\geq \varepsilon $
for any
$\beta \in \mathbb {B}_{F}$
. Put
$w=g^{-1}\varepsilon $
. Let
$\mathcal {V}$
be the admissible affinoid open subset of
$\mathrm {sp}^{-1}\left (\bar {P}\right )$
defined as
$$ \begin{align*} \mathcal{V}=\left\{\left(t_{\beta}\right)_{\beta\in \mathbb{B}_{F}}\in \prod_{\beta\in\mathbb{B}_{F}} \mathcal{A}_{\beta}\left[0,p^{-w}\right]\ \middle|\ \prod_{\beta\in\tau\left(\bar{P}\right)}\left\lvert t_{\beta}\right\rvert\geq p^{-v}\right\}. \end{align*} $$
Since
$P\in \mathcal {V}\subseteq \mathcal {X}_{v}$
and
$\mathcal {V}\cap \mathcal {X}_{\varepsilon }\neq \emptyset $
, by an argument as before we reduce ourselves to showing that
$\mathcal {V}$
is connected.
For this, replacing K by its finite extension, we may assume that there exist
$a,b\in \mathcal {O}_{K}$
satisfying
$v_{p}(a)=v$
and
$v_{p}(b)=w$
. Consider the ring
$$ \begin{align*} \mathfrak{R}=\mathcal{O}_{K}\left\langle T_{\beta}, U_{\beta}\mid \beta\in \mathbb{B}_{F}\right\rangle\langle T\rangle/\left(Tf-a,b U_{\beta}-T_{\beta}\mid \beta\in \mathbb{B}_{F}\right) \end{align*} $$
with
$f=\prod _{\beta \in \tau \left (\bar {P}\right )}T_{\beta }$
. Then
$\mathcal {V}=\mathrm {Sp}(\mathfrak {R}[1/p])$
and the special fiber
$$ \begin{align*} \mathrm{Spec}\left(k\left[T_{\beta}, U_{\beta}\mid \beta\in \mathbb{B}_{F}\right][T]/\left(Tf,T_{\beta}\mid \beta\in \mathbb{B}_{F}\right)\right)=\mathrm{Spec}\left(k\left[U_{\beta}\mid \beta\in \mathbb{B}_{F}\right][T]\right) \end{align*} $$
is reduced and connected. Then Lemma 3.2 concludes the proof.
3.1.4 Canonical subgroups over moduli spaces
Let n be a positive integer. Let
$\underline {v}=\left (v_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
be a g-tuple satisfying
$$ \begin{align*} v_{\beta}\in [0,(p-1)/p^{n})\cap \mathbb{Q} \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
. Note that the
$1/(p^{n}(p-1))$
th lower ramification subgroups can be patched into a rigid analytic family [Reference Hattori28, Lemma 5.6]. Let R be an object of
$\mathbf {NAdm}$
and put
$\mathcal {U}=\mathrm {Sp}(R[1/p])$
. Let
$\mathcal {U}\to \mathfrak {M}(\mu _{N}, \mathfrak {c})\left (\underline {v}\right )^{\mathrm {rig}}$
be any morphism of rigid analytic varieties over K. This defines an HBAV
$\bar {A}^{\mathrm {un}}\rvert _{R}$
over
$\mathrm {Spec}(R)$
. For any rig-point
$x\in \mathrm {Spec}(R)$
, we have the canonical subgroup
$\mathcal {C}_{n}\left (\left (\bar {A}^{\mathrm {un}}\rvert _{R}\right )_{x}\right )$
by [Reference Hattori29, Theorem 8.1]. [Reference Hattori29, Theorem 8.1(7)] implies that they can be patched into an admissible open
$\mathcal {O}_{F}$
-submodule
$C_{n}$
of
$\bar {\mathcal {A}}^{\mathrm {un}}[p^{n}]\rvert _{\mathcal {U}}$
. By [Reference Andreatta, Iovita and Pilloni3, Proposition 4.1.3], it uniquely extends to a finite flat subgroup scheme
$\mathcal {C}_{n}$
of
$\bar {A}^{\mathrm {un}}\rvert _{R}$
over
$ \mathrm {Spec}(R)$
. Since
$\mathcal {C}_{n}$
agrees with the scheme-theoretic closure of
$C_{n}$
in
$\left (\bar {A}^{\mathrm {un}}\rvert _{R}\right )[p^{n}]$
, we see that
$\mathcal {C}_{n}$
is stable under the
$\mathcal {O}_{F}$
-action.
On the other hand, on a formal open neighborhood
$\mathfrak {U}$
of a point of the boundary satisfying
$\mathfrak {U}\subseteq \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
, the unit component
$\bar {\mathfrak {A}}^{\mathrm {un}}[p^{n}]^{0}\rvert _{\mathfrak {U}}$
is quasi-finite and flat over
$\mathfrak {U}$
with constant degree on each fiber by [Reference Rapoport36, p. 297(ii)]. Thus it is finite and flat. Then by gluing along
$\mathfrak {M}(\mu _{N},\mathfrak {c})(0)$
, we obtain a finite flat formal subgroup scheme
$\mathcal {C}_{n}$
of
$\bar {\mathfrak {A}}^{\mathrm {un}}$
over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
and its generic fiber
$C_{n}$
, which we refer to as the canonical subgroup of level n.
Let R be a topological
$\mathcal {O}_{K}$
-algebra which is quasi-idyllic with respect to the p-adic topology [Reference Abbes1, 1.10.1.1]. Since any finitely generated R-module is automatically p-adically complete [Reference Abbes1, Proposition 1.10.2], any finitely presented flat formal group scheme over
$\mathrm {Spf}(R)$
can be identified with a finitely presented flat group scheme over
$\mathrm {Spec}(R)$
. Thus we have a theory of Cartier duality for any finitely presented flat formal group scheme
$\mathcal {G}$
over any quasi-idyllic p-adic formal scheme, and we can define the Hodge–Tate map
$$ \begin{align*} \mathrm{HT}_{\mathcal{G}}:\mathcal{G}(R)\to \omega_{\mathcal{G}^{\vee}},\quad x\mapsto x^{*}\left(\frac{dT}{T}\right). \end{align*} $$
From the construction, we see that the restriction of the Cartier dual
$\mathcal {C}_{n}^{\vee }\rvert _{\bar {\mathfrak {M}}\left (\mu _{N},\mathfrak {c}\right )(0)}$
to the ordinary locus is finite and étale.
Note that for the function
$$ \begin{align*} \delta_{K}(v)=\min\left([v/(p-1),+\infty)\cap e^{-1}\mathbb{Z}\right), \end{align*} $$
the ideal
$m_{K}^{\geqslant v/(p-1)}$
is generated by
$\pi _{K}^{e\delta _{K}(v)}$
. Then we have the following variant of [Reference Andreatta, Iovita and Pilloni3, Propositions 4.2.1 and 4.2.2]:
Lemma 3.5. Let
$\underline {v}=\left (v_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
be a g-tuple of nonnegative rational numbers satisfying
$$ \begin{align*} v:=\max\left\{v_{\beta}\mid \beta\in \mathbb{B}_{F}\right\}< (p-1)/p^{n}. \end{align*} $$
Let R be an object of
$\mathbf {NAdm}$
. For any morphism of admissible formal schemes
$\hat {f}:\mathrm {Spf}(R)\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
over
$\mathcal {O}_{K}$
, consider the pullback
$G=\bar {A}^{\mathrm {un}}\rvert _{R}$
by the unique algebraization
$\mathrm {Spec}(R)\to \bar {M}(\mu _{N},\mathfrak {c})$
of
$\hat {f}$
and
$\mathcal {H}_{n}=\mathcal {C}_{n}\rvert _{\mathrm {Spf}(R)}$
, which is a subgroup scheme of the formal completion of G.
-
(1) For any rational number
$i\in e^{-1}\mathbb {Z}_{\geq 0}$
satisfying
$i\leq n-v(p^{n}-1)/(p-1)$
, the natural map
$\omega _{G}\otimes _{\mathcal {O}_{K}} \mathcal {O}_{K,i}\to \omega _{\mathcal {H}_{n}} \otimes _{\mathcal {O}_{K}} \mathcal {O}_{K,i}$
is an isomorphism. -
(2) Assume that we have an isomorphism of
$\mathcal {O}_{F}$
-modules
$\mathcal {H}_{n}^{\vee }(R)\simeq \mathcal {O}_{F} /p^{n}\mathcal {O}_{F}$
. If there exists a rational number
$i\in e^{-1}\mathbb {Z}_{\geq 0}$
satisfying
$\delta _{K}(v)<i\leq n-v(p^{n}-1)/(p-1)$
, then the cokernel of the linearization of the Hodge–Tate map is killed by
$$ \begin{align*} \mathrm{HT}_{\mathcal{H}^{\vee}_{n}}\otimes 1: \mathcal{H}_{n}^{\vee}(R)\otimes R\to \omega_{\mathcal{H}_{n}} \end{align*} $$
$m_{K}^{\geqslant v/(p-1)}$
.
Proof. Since the ordinary case is trivial, by a gluing argument we may assume that
$\hat {f}$
factors through
$\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
. By replacing
$\mathrm {Spf}(R)$
with its formal affine open subscheme, we may assume that R is an integral domain and
$\omega _{G}$
is a free
$\mathcal {O}_{F}\otimes R$
-module of rank
$1$
. The first assertion follows by reducing it to [Reference Hattori29, Theorem 8.1(8)] in the same way as [Reference Andreatta, Iovita and Pilloni3, Proposition 4.2.1]. For the second assertion, take surjections
$R^{g} \to \mathcal {H}_{n}^{\vee }(R)\otimes R \simeq (R/p^{n} R)^{g}$
and
$R^{g}\simeq \omega _{G}\to \omega _{\mathcal {H}_{n}}$
. Then the map
$\mathrm {HT}_{\mathcal {H}^{\vee }_{n}} \otimes 1$
can be identified with the reduction of the map defined by some matrix
$\gamma \in M_{g}(R)$
. It suffices to show
$m_{K}^{\geqslant v/(p-1)} R^{g}\subseteq \gamma (R^{g})$
. Let
$\mathfrak {p}$
be a prime ideal of R of height
$1$
and let
$\hat {R}_{\mathfrak {p}}$
be the completion of the local ring
$R_{\mathfrak {p}}$
. [Reference Hattori29, Theorem 8.1(9)] implies
$m_{K}^{\geqslant v/(p-1)} \hat {R}_{\mathfrak {p}}^{g}\subseteq \gamma \left (\hat {R}_{\mathfrak {p}}^{g}\right )+\pi _{K}^{e i}\hat {R}_{\mathfrak {p}}^{g}$
and thus
$m_{K}^{\geqslant v/(p-1)} \hat {R}_{\mathfrak {p}}^{g}\subseteq \gamma \left (\hat {R}_{\mathfrak {p}}^{g}\right )$
. This shows
$m_{K}^{\geqslant v/(p-1)} R_{\mathfrak {p}}^{g}\subseteq \gamma \left (R_{\mathfrak {p}}^{g}\right )$
and
$\det (\gamma )\neq 0$
. Since R is normal,
$\gamma (R^{g})$
is the intersection of
$\gamma \left (R_{\mathfrak {p}}^{g}\right )$
for every such
$\mathfrak {p}$
, and the assertion follows.
3.2 Connected neighborhoods of critical points
Let
$Y_{\mathfrak {c},p}$
be the moduli scheme parametrizing the isomorphism classes of pairs
$(A, \mathcal {H})$
over schemes
$S/\mathrm {Spec}(\mathcal {O}_{K})$
, where A is an HBAV over S with
$\mathfrak {c}$
-polarization and
$\Gamma _{00}(N)$
-structure, and
$\mathcal {H}$
is a finite locally free closed
$\mathcal {O}_{F}$
-subgroup scheme of
$A[p]$
of rank
$p^{g}$
over S such that
$\mathcal {H}$
is isotropic in the sense of [Reference Goren and Kassaei27, §2.1]. Then
$Y_{\mathfrak {c},p}$
is projective over
$M(\mu _{N},\mathfrak {c})$
[Reference Stamm38, p. 415]. For
$S=\mathrm {Spec}(\mathcal {O}_{L})$
with some extension
$L/K$
of complete valuation fields and an ideal
$\mathfrak {a}$
of
$\mathcal {O}_{F}$
, we say that
$\mathcal {H}$
is
$\mathfrak {a}$
-cyclic if the
$\mathcal {O}_{F}$
-module
$\mathcal {H}\left (\mathcal {O}_{\bar {L}}\right )$
is isomorphic to
$\mathcal {O}_{F}/\mathfrak {a}$
, where
$\bar {L}$
is an algebraic closure of L. Then
$\mathcal {H}$
is isotropic in this sense if and only if
$\mathcal {H}$
is p-cyclic.
Let
$\mathfrak {Y}_{\mathfrak {c},p}$
be the p-adic formal completion of
$Y_{\mathfrak {c},p}$
and let
$\mathcal {Y}_{\mathfrak {c},p}$
be its Raynaud generic fiber. Note that they are separated. By [Reference Rapoport36, Lemme 3.1], we have
$\mathcal {Y}_{\mathfrak {c},p}(L)=\mathfrak {Y}_{\mathfrak {c},p}(\mathcal {O}_{L})=Y_{\mathfrak {c},p}(\mathcal {O}_{L})$
for any extension
$L/K$
of complete discrete valuation fields. In this subsection, we construct a connected admissible affinoid open neighborhood of a point
$Q=[(A,\mathcal {H})]$
of
$\mathcal {Y}_{\mathfrak {c},p}$
satisfying
$\mathrm {Hdg}_{\beta }(A)=p/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
inside the base extension
$\mathcal {Y}_{\mathfrak {c},p,\mathbb {C}_{p}}=\mathcal {Y}_{\mathfrak {c},p}\hat {\otimes }_{K} \mathbb {C}_{p}$
, assuming
$f_{\mathfrak {p}}\leq 2$
for any
$\mathfrak {p}\mid p$
.
Lemma 3.6. There exists a point of
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$
corresponding to an HBAV A over the ring of integers
$\mathcal {O}_{L}$
of a finite extension
$L/K$
satisfying
$ \mathrm {Hdg}_{\beta }(A)=p/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
.
Proof. Consider the stratum
$W_{\mathbb {B}_{F}}$
of the special fiber
$M(\mu _{N},\mathfrak {c})_{k}$
as in [Reference Goren and Kassaei27, §2.5]. Since
$W_{\mathbb {B}_{F}}$
is nonempty, there exists a point
$P\in \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$
such that
$\bar {P}=\mathrm {sp}(P)\in W_{\mathbb {B}_{F}}$
for the specialization map
$\mathrm {sp}: \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\to \bar {M}(\mu _{N},\mathfrak {c})_{k}$
as before. Since
$\tau (\bar {P})=\mathbb {B}_{F}$
, the identifications (3.2) and (3.3) yield the lemma.
Proposition 3.7. Suppose
$f_{\mathfrak {p}}\leq 2$
for any
$\mathfrak {p}\mid p$
. Let
$L/K$
be a finite extension in
$\bar {\mathbb {Q}}_{p}$
and let l be the residue field of L. Let
$K^{\prime }$
be the composite field of K and
$\mathrm {Frac}(W(l))$
in
$\bar {\mathbb {Q}}_{p}$
. Let
$[(A,\mathcal {H})]$
be an element of
$Y_{\mathfrak {c},p}(\mathcal {O}_{L})$
satisfying
$\mathrm {Hdg}_{\beta }(A)=p/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
, and let Q be the element of
$\mathcal {Y}_{\mathfrak {c},p}(L)$
it defines. Let
$$ \begin{align*} \mathrm{sp}:\mathcal{Y}_{\mathfrak{c},p}\to \left(Y_{\mathfrak{c},p}\right)_{k}=Y_{\mathfrak{c},p}\times_{\mathcal{O}_{K}}\mathrm{Spec}(k) \end{align*} $$
be the specialization map with respect to
$\mathfrak {Y}_{\mathfrak {c},p}$
and put
$\bar {Q}=\mathrm {sp}(Q)$
. We define
$$ \begin{align*} &\mathcal{V}_{\mathfrak{c},Q}=\left\{Q^{\prime}=[(A^{\prime},\mathcal{H}^{\prime})] \in \mathrm{sp}^{-1}\left(\bar{Q}\right) \ \middle|\ \frac{1}{p+1}\leq\deg_{\beta}(A'[p]/\mathcal{H}^{\prime})\leq \frac{p}{p+1},\right. \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\mathrm{Hdg}_{\beta}(A^{\prime})\leq \frac{p}{p+1}\text{ for any }\beta\in \mathbb{B}_{F}\right\}, \end{align*} $$
$$ \begin{align*} \mathcal{V}_{\mathfrak{c},Q}\left(\tfrac{1}{p+1}\right)=\left\{Q^{\prime}=[(A^{\prime},\mathcal{H}^{\prime})] \in \mathcal{V}_{\mathfrak{c},Q}\ \middle|\ \deg_{\beta}(A^{\prime}[p]/\mathcal{H}^{\prime})\leq \frac{1}{p+1}\text{ for any }\beta\in \mathbb{B}_{F}\right\}. \end{align*} $$
Then they are admissible affinoid open subsets of
$\mathcal {Y}_{\mathfrak {c},p}$
defined over
$K^{\prime }$
such that
$\mathcal {V}_{\mathfrak {c},Q}\hat {\otimes }_{K^{\prime }}\mathbb {C}_{p}$
is connected.
Proof. By the assumption
$f_{\mathfrak {p}}\leq 2$
and [Reference Hattori29, Proposition 6.1], we have the equality
$\deg _{\beta }(A[p]/\mathcal {H})=p/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
. [Reference Tian39, Proposition 4.2] shows that this value is equal to the one denoted by
$\nu _{\beta }(Q)$
in [Reference Goren and Kassaei27, §4.2]. In particular, the definition of
$\nu _{\beta }(Q)$
in [Reference Goren and Kassaei27, §4.2] implies
$I\left (\bar {Q}\right )=\mathbb {B}_{F}$
with the notation of [Reference Goren and Kassaei27, (2.3.2)].
We claim that the complete local ring
$\hat {\mathcal {O}}_{Y_{\mathfrak {c},p},\bar {Q}}$
of
$Y_{\mathfrak {c},p}$
at
$\bar {Q}$
is isomorphic to the ring
$$ \begin{align} \mathfrak{B}^{\prime}=\mathcal{O}_{K^{\prime}}\left[\left[X_{\beta},Y_{\beta}\mid \beta\in \mathbb{B}_{F}\right]\right]/\left(X_{\beta} Y_{\beta}-p\mid \beta\in \mathbb{B}_{F}\right) \end{align} $$
and that there exists
$g_{\beta }\in (\mathfrak {B}^{\prime })^{\times }$
such that for any finite extension
$E/K^{\prime }$
and any
$\mathcal {O}_{K^{\prime }}$
-algebra homomorphism
$x: \mathfrak {B}^{\prime }\to \mathcal {O}_{E}$
, the corresponding
$\mathcal {O}_{E}$
-valued point
$[(A^{\prime },\mathcal {H}^{\prime })]$
of
$Y_{\mathfrak {c},p}$
satisfies
$$ \begin{align*} \deg_{\beta}(A^{\prime}[p]/\mathcal{H}^{\prime})=\min\{v_p(X_\beta(x)),1\},\qquad \mathrm{Hdg}_{\beta}(A^{\prime})=\min\{v_p((X_\beta+g_\beta Y^p_{\sigma^{-1}\circ\beta})(x)),1\}. \end{align*} $$
Indeed, let
$Y_{\mathfrak {c}}$
be a moduli scheme over W similar to
$Y_{\mathfrak {c},p}$
considered in [Reference Goren and Kassaei27, §2.1]. Let R be the affine algebra of an affine open neighborhood of
$\bar {Q}$
in
$Y_{\mathfrak {c}}$
and let
$m_{\bar {Q}}$
be the maximal ideal of R corresponding to
$\bar {Q}$
. The ring
$\hat {\mathcal {O}}_{Y_{\mathfrak {c},p},\bar {Q}}$
is equal to the completion of the local ring of
$R\otimes _{W} \mathcal {O}_{K}$
at the kernel
$n_{\bar {Q}}$
of the map
$R\otimes _{W} \mathcal {O}_{K} \to l$
associated to
$m_{\bar {Q}}$
.
By Stamm’s theorem [Reference Stamm38] (see also [Reference Goren and Kassaei27, Theorem 2.4.1′]), the
$m_{\bar {Q}}$
-adic completion
$\hat {R}_{m_{\bar {Q}}}$
of the localization
$R_{m_{\bar {Q}}}$
is isomorphic to the ring
$$ \begin{align*} \mathfrak{B}=W(l)\left[\left[X_{\beta},Y_{\beta}\mid \beta\in \mathbb{B}_{F}\right]\right]/\left(X_{\beta} Y_{\beta}-p\mid \beta\in \mathbb{B}_{F}\right). \end{align*} $$
Moreover, since
$\mathrm {Hdg}_{\beta }(A)\neq 0$
for any
$\beta \in \mathbb {B}_{F}$
, equation (3.3) implies
$\tau \left (\bar {Q}\right )=\mathbb {B}_{F}$
. Thus, for any finite extension
$E/\mathrm {Frac}(W(l))$
and any
$W(l)$
-algebra homomorphism
$x:\mathfrak {B}\to \mathcal {O}_{E}$
, the corresponding HBAV
$A^{\prime }$
satisfies
$v\left (t_{\beta }(x)\right )=\mathrm {Hdg}_{\beta }(A^{\prime })$
. By [Reference Goren and Kassaei27, Lemma 2.8.1] and the definition of
$\nu _{\beta }(Q)$
in [Reference Goren and Kassaei27, §4.2], the isomorphism
$\hat {R}_{m_{\bar {Q}}}\simeq \mathfrak {B}$
gives an identification of
$\deg _{\beta }$
and
$\mathrm {Hdg}_{\beta }$
for the ring
$\mathfrak {B}$
as claimed before. Then the claim follows from Lemma 3.3.
By [Reference de Jong19, Lemma 7.2.5], we have
$$ \begin{align*} \mathrm{sp}^{-1}\left(\bar{Q}\right)=(\mathrm{Spf}(\mathfrak{B}^{\prime}))^{\mathrm{rig}}. \end{align*} $$
Thus
$\mathcal {V}_{\mathfrak {c},Q}$
is the
$K^{\prime }$
-affinoid variety whose affinoid ring is the quotient of the Tate algebra
$$ \begin{align*} K^{\prime} \left\langle X_{\beta}, Y_{\beta}, U_{\beta}, V_{\beta}, W_{\beta} \mid \beta\in \mathbb{B}_{F} \right\rangle \end{align*} $$
by the ideal generated by
$$ \begin{align*} X_{\beta}^{p+1}-p U_{\beta},\quad X_{\beta}^{p+1} V_{\beta} - p^{p},\quad X_{\beta} Y_{\beta}-p,\quad W_{\beta}\left(X_{\beta} +g_{\beta} Y_{\sigma^{-1}\circ \beta}^{p}\right)^{p+1} - p^{p} \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
. From this, we also obtain a similar description of
$\mathcal {V}_{\mathfrak {c},Q}\left (\tfrac {1}{p+1}\right )$
as a
$K^{\prime }$
-affinoid variety.
Next we prove that the base extension
$\mathcal {V}_{\mathfrak {c},Q}\hat {\otimes }_{K^{\prime }} \mathbb {C}_{p}$
is connected. Put
$r=1/(p+1)$
and
$s=p/(p+1)$
. Fix a
$(p+1)$
th root
$\varpi =p^{1/ (p+1)}$
of p in
$\bar {\mathbb {Q}}_{p}$
. Then the affinoid ring
$B_{Q,\mathbb {C}_{p}}$
of
$\mathcal {V}_{\mathfrak {c},Q}\hat {\otimes }_{K^{\prime }} \mathbb {C}_{p}$
is also isomorphic to the quotient of the Tate algebra
$$ \begin{align*} \mathbb{C}_{p} \left\langle X_{\beta}, Y_{\beta}, U_{\beta}, V_{\beta}, W_{\beta} \mid \beta\in \mathbb{B}_{F} \right\rangle \end{align*} $$
by the ideal generated by
$$ \begin{align*} X_{\beta}-\varpi U_{\beta},\quad X_{\beta} V_{\beta} - \varpi^{p},\quad X_{\beta} Y_{\beta}-\varpi^{p+1},\quad W_{\beta}\left(X_{\beta} +g_{\beta} Y_{\sigma^{-1}\circ \beta}^{p}\right) - \varpi^{p} \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
. Note that in the ring
$B_{Q,\mathbb {C}_{p}}$
we also have
$Y_{\beta }-\varpi V_{\beta }=0$
. Hence
$B_{Q,\mathbb {C}_{p}}$
is isomorphic to the quotient of the ring
$$ \begin{align*} \mathbb{C}_{p} \left\langle U_{\beta}, V_{\beta}, W_{\beta} \mid \beta\in \mathbb{B}_{F} \right\rangle \end{align*} $$
by the ideal generated by
$$ \begin{align*} U_{\beta} V_{\beta} -\varpi^{p-1},\quad F_{\beta}:=W_{\beta}\left(U_{\beta}+\varpi^{p-1} g^{\prime}_{\beta} V_{\sigma^{-1}\circ \beta}^{p}\right)-\varpi^{p-1} \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
with some
$g^{\prime }_{\beta }\in \mathfrak {A}_{Q,\mathbb {C}_{p}}^{\times }$
, where
$$ \begin{align*} \mathfrak{A}_{Q,\mathbb{C}_{p}}=\mathcal{O}_{\mathbb{C}_{p}}\left\langle U_{\beta},V_{\beta}\mid \beta\in \mathbb{B}_{F}\right\rangle/\left(U_{\beta} V_{\beta}-\varpi^{p-1}\mid \beta\in \mathbb{B}_{F}\right). \end{align*} $$
From these equations, we see that
$$ \begin{align*} G_{\beta}:=V_{\beta}-W_{\beta}\left(1+g^{\prime}_{\beta} V_{\beta} V_{\sigma^{-1}\circ \beta}^{p}\right)=0 \end{align*} $$
in this quotient. Since
$$ \begin{align*} F_{\beta}\equiv -U_{\beta} G_{\beta} \bmod U_{\beta} V_{\beta}-\varpi^{p-1}, \end{align*} $$
we obtain
$$ \begin{align*} B_{Q,\mathbb{C}_{p}}\simeq \mathbb{C}_{p}\left\langle U_{\beta}, V_{\beta}, W_{\beta}\mid \beta\in \mathbb{B}_{F}\right\rangle/\left(U_{\beta} V_{\beta}-\varpi^{p-1}, G_{\beta}\mid \beta\in \mathbb{B}_{F}\right). \end{align*} $$
Note that the ring
$$ \begin{align*} \mathfrak{B}_{Q,\mathbb{C}_{p}}=\mathcal{O}_{\mathbb{C}_{p}}\left\langle U_{\beta}, V_{\beta}, W_{\beta} \mid \beta\in \mathbb{B}_{F}\right\rangle/\left(U_{\beta} V_{\beta}-\varpi^{p-1}, G_{\beta}\mid \beta\in \mathbb{B}_{F}\right) \end{align*} $$
is a flat
$\mathcal {O}_{\mathbb {C}_{p}}$
-algebra. Indeed, consider the polynomial ring
$\mathfrak {A}_{Q,\mathbb {C}_{p}}\left [W_{\beta }\right ]$
. Since the coefficients of
$G_{\beta }$
as a polynomial of
$W_{\beta }$
generate the unit ideal
$\mathfrak {A}_{Q,\mathbb {C}_{p}}$
, by a limit argument reducing to the Noetherian case and using [Reference Matsumura34, (20.F), Corollary 2] we see that the
$\mathfrak {A}_{Q,\mathbb {C}_{p}}$
-algebra
$$ \begin{align*} \mathfrak{A}_{Q,\mathbb{C}_{p}}\left[W_{\beta}\mid \beta\in \mathbb{B}_{F}\right]/\left(G_{\beta}\mid \beta\in \mathbb{B}_{F}\right) \end{align*} $$
is flat. By [Reference Abbes1, Proposition 1.10.2(ii)], the p-adic completion of this algebra is
$\mathfrak {B}_{Q,\mathbb {C}_{p}}$
. Since the
$\mathcal {O}_{\mathbb {C}_{p}}$
-algebra
$\mathfrak {A}_{Q,\mathbb {C}_{p}}$
is flat, the p-adic completion
$\mathfrak {B}_{Q,\mathbb {C}_{p}}$
is also flat over
$\mathcal {O}_{\mathbb {C}_{p}}$
.
Put
$\bar {G}_{\beta }=G_{\beta } \bmod m_{\mathbb {C}_{p}}$
and
$$ \begin{align*} \bar{R}=\bar{\mathbb{F}}_{p} \left[U_{\beta}, V_{\beta}, W_{\beta}\mid \beta\in \mathbb{B}_{F}\right],\qquad \bar{J}=\left(U_{\beta} V_{\beta}, \bar{G}_{\beta}\mid \beta\in \mathbb{B}_{F}\right). \end{align*} $$
Next we claim that the reduction
$\bar {\mathfrak {B}}_{Q,\mathbb {C}_{p}}=\bar {R}/\bar {J}$
of
$\mathfrak {B}_{Q,\mathbb {C}_{p}}$
is reduced and
$\mathrm {Spec}\left (\bar {\mathfrak {B}}_{Q,\mathbb {C}_{p}}\right )$
is connected. For reducedness, it suffices to show that the localization at every maximal ideal is reduced. Let
$\mathfrak {M}$
be any maximal ideal of
$\bar {R}$
containing
$\bar {J}$
. Then we have
$$ \begin{align*} 1+g^{\prime}_{\beta} V_{\beta} V_{\sigma^{-1}\circ \beta}^{p} \notin \mathfrak{M}, \end{align*} $$
since, supposing the contrary,
$\bar {G}_{\beta }\in \mathfrak {M}$
implies
$V_{\beta }\in \mathfrak {M}$
and
$1\in \mathfrak {M}$
, which is a contradiction. Thus in the ring
$\bar {R}_{\mathfrak {M}}$
we have
$$ \begin{align*} W_{\beta}-V_{\beta}\left(1+g^{\prime}_{\beta} V_{\beta} V_{\sigma^{-1}\circ \beta}^{p}\right)^{-1}\in \bar{J} \bar{R}_{\mathfrak{M}} \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
. Hence the localization
$\left (\bar {\mathfrak {B}}_{Q,\mathbb {C}_{p}}\right )_{\mathfrak {M}}$
is isomorphic to the localization of the ring
$$ \begin{align*} \bar{\mathbb{F}}_{p}\left[U_{\beta}, V_{\beta}\mid \beta\in \mathbb{B}_{F}\right]/\left(U_{\beta} V_{\beta}\mid \beta\in \mathbb{B}_{F}\right) \end{align*} $$
at the pullback of
$\mathfrak {M}$
, which is reduced.
Let us show the connectedness. Let
$\mathbb {B}_{F}=\mathbb {B}_{U} \coprod \mathbb {B}_{V}$
be a decomposition into the disjoint union of two subsets. Consider the closed subscheme
$F_{\mathbb {B}_{U},\mathbb {B}_{V}}$
of
$\mathrm {Spec}\left (\bar {\mathfrak {B}}_{Q,\mathbb {C}_{p}}\right )$
defined by
$U_{\beta }=0$
for
$\beta \in \mathbb {B}_{U}$
and
$V_{\beta }=0$
for
$\beta \in \mathbb {B}_{V}$
. Since every
$F_{\mathbb {B}_{U},\mathbb {B}_{V}}$
contains the point defined by
$U_{\beta }=V_{\beta }=W_{\beta }=0$
for any
$\beta \in \mathbb {B}_{F}$
, it is enough to show that
$F_{\mathbb {B}_{U}, \mathbb {B}_{V}}$
is connected for any such decomposition of
$\mathbb {B}_{F}$
. Put
$$ \begin{align*} \bar{\mathfrak{A}}_{\mathbb{B}_{U},\mathbb{B}_{V}}=\bar{\mathbb{F}}_{p}\left[U_{\beta}, V_{\beta}\mid \beta\in\mathbb{B}_{F}\right]/\left(U_{\beta}\ (\beta\in \mathbb{B}_{U}), V_{\beta}\ (\beta\in\mathbb{B}_{V})\right). \end{align*} $$
Note that the
$\bar {\mathfrak {A}}_{\mathbb {B}_{U},\mathbb {B}_{V}}$
-algebra
$$ \begin{align*} \bar{\mathfrak{A}}_{\mathbb{B}_{U},\mathbb{B}_{V}}\left[W_{\beta}\mid \beta\in \mathbb{B}_{F}\right]/\left(\bar{G}_{\beta}\mid \beta\in \mathbb{B}_{F}\right) \end{align*} $$
is flat. From this we see that the affine algebra of
$F_{\mathbb {B}_{U},\mathbb {B}_{V}}$
can be identified with the subring
$$ \begin{align*} \bar{\mathfrak{A}}_{\mathbb{B}_{U},\mathbb{B}_{V}} \left[\frac{V_{\beta}}{1+g^{\prime}_{\beta} V_{\beta} V_{\sigma^{-1}\circ\beta}}\ \middle|\ \beta\in \mathbb{B}_{F}\right] \end{align*} $$
of
$\mathrm {Frac}\left (\bar {\mathfrak {A}}_{\mathbb {B}_{U},\mathbb {B}_{V}}\right )$
, which is an integral domain. Hence we obtain the connectedness of
$\bar {\mathfrak {B}}_{Q,\mathbb {C}_{p}}$
. By [Reference de Jong19, Lemma 7.1.9],
$\mathrm {sp}^{-1}\left (\bar {Q}\right )$
is reduced, and [Reference Conrad17, Lemma 3.3.1(1)] shows that
$\mathcal {V}_{\mathfrak {c},Q}\hat {\otimes }_{K^{\prime }} \mathbb {C}_{p}$
is also reduced. Then Lemma 3.2 shows that
$\mathcal {V}_{\mathfrak {c},Q}\hat {\otimes }_{K^{\prime }}\mathbb {C}_{p}$
is connected.
Lemma 3.8. Suppose
$f_{\mathfrak {p}}\leq 2$
for any
$\mathfrak {p}\mid p$
. Let
$L/K$
be a finite extension. Let
$[(A,\mathcal {H})]$
be an element of
$Y_{\mathfrak {c},p}(\mathcal {O}_{L})$
satisfying
$$ \begin{align*} \deg_{\beta}(A[p]/\mathcal{H})\leq p/(p+1),\qquad \mathrm{Hdg}_{\beta}(A)\leq p/(p+1) \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
. Then for any
$\mathfrak {p}\mid p$
, either
$A[p]_{\mathfrak {p}}$
has the canonical subgroup of level
$1$
which is equal to
$\mathcal {H}_{\mathfrak {p}}$
, or
$\mathrm {Hdg}_{\beta }(A)=p/(p+1)$
for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
.
Proof. Suppose
$\mathrm {Hdg}_{\beta _{0}}(A)<p/(p+1)$
for some
$\beta _{0}\in \mathbb {B}_{\mathfrak {p}}$
. Since we have
$\mathrm {Hdg}_{\beta }(A)\leq p/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
, the assumption on
$f_{\mathfrak {p}}$
implies that the inequality
$$ \begin{align*} \mathrm{Hdg}_{\beta}(A)+ p \mathrm{Hdg}_{\sigma^{-1}\circ\beta}(A)<p \end{align*} $$
holds for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
. By [Reference Hattori29, Theorem 4.1], the
$\mathcal {O}_{F_{\mathfrak {p}}}$
-
$\mathrm {ADBT}_{1}\ A[p]_{\mathfrak {p}}$
has the canonical subgroup
$\mathcal {C}_{\mathfrak {p}}$
.
Suppose
$\mathcal {H}_{\mathfrak {p}}\neq \mathcal {C}_{\mathfrak {p}}$
. For any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
, [Reference Hattori29, Corollary 5.3(1)] implies that
$$ \begin{align*} \mathrm{Hdg}_{\beta}\left(p^{-1}\mathcal{H}_{\mathfrak{p}}/\mathcal{H}_{\mathfrak{p}}\right)=p^{-1}\mathrm{Hdg}_{\sigma\circ\beta}\left(A[p]_{\mathfrak{p}}\right)=p^{-1}\mathrm{Hdg}_{\sigma\circ\beta}(A[p])\leq 1/(p+1) \end{align*} $$
and that
$A[p]_{\mathfrak {p}}/\mathcal {H}_{\mathfrak {p}}$
is the canonical subgroup of
$p^{-1}\mathcal {H}_{\mathfrak {p}}/\mathcal {H}_{\mathfrak {p}}$
. Thus we have
$$ \begin{align*} \deg_{\beta}(A[p]/\mathcal{H})=\deg_{\beta}\left(A[p]_{\mathfrak{p}}/\mathcal{H}_{\mathfrak{p}}\right)=1-\mathrm{Hdg}_{\beta}\left(p^{-1}\mathcal{H}_{\mathfrak{p}}/\mathcal{H}_{\mathfrak{p}}\right)\geq p/(p+1), \end{align*} $$
which yields
$\deg _{\beta }(A[p]/\mathcal {H})=p/(p+1)$
and
$\mathrm {Hdg}_{\beta }(A[p])=p/(p+1)$
for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
. This contradicts the choice of
$\beta _{0}$
.
Corollary 3.9. Suppose
$f_{\mathfrak {p}}\leq 2$
for any
$\mathfrak {p}\mid p$
. Let
$L/K$
be a finite extension. Let
$[(A^{\prime },\mathcal {H}^{\prime })]$
be an element of
$Y_{\mathfrak {c},p}(\mathcal {O}_{L})$
such that
$\left [\left (A^{\prime }_{L},\mathcal {H}^{\prime }_{L}\right )\right ]\in \mathcal {V}_{\mathfrak {c},Q}(L)$
. Then for any finite flat closed p-cyclic
$\mathcal {O}_{F}$
-subgroup scheme
$\mathcal {D}$
of
$A^{\prime }[p]$
over
$\mathcal {O}_{L}$
satisfying
$\mathcal {D}_{L}\cap \mathcal {H}^{\prime }_{L}=0$
, we have
$$ \begin{align*} \mathrm{Hdg}_{\beta}(A^{\prime}/\mathcal{D})\leq 1/(p+1) \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
, and
$A^{\prime }[p]/\mathcal {D}$
is the canonical subgroup of
$A^{\prime }/\mathcal {D}$
of level
$1$
.
Proof. Write
$\mathcal {D}=\bigoplus _{\mathfrak {p}\mid p} \mathcal {D}_{\mathfrak {p}}$
. The assumption implies
$\mathcal {D}_{\mathfrak {p}}\neq \mathcal {H}^{\prime }_{\mathfrak {p}}$
for any
$\mathfrak {p}\mid p$
. If
$\mathcal {H}^{\prime }_{\mathfrak {p}}$
is the canonical subgroup of
$A^{\prime }[p]_{\mathfrak {p}}$
, then [Reference Hattori29, Corollary 5.3(1)] implies that
$$ \begin{align*} \mathrm{Hdg}_{\beta}(A^{\prime}/\mathcal{D})=\mathrm{Hdg}_{\beta}\left(p^{-1}\mathcal{D}_{\mathfrak{p}}/\mathcal{D}_{\mathfrak{p}}\right)=p^{-1}\mathrm{Hdg}_{\sigma\circ\beta}(A^{\prime}[p])\leq 1/(p+1) \end{align*} $$
for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
and that
$A^{\prime }[p]_{\mathfrak {p}}/\mathcal {D}_{\mathfrak {p}}$
is the canonical subgroup of
$p^{-1}\mathcal {D}_{\mathfrak {p}}/\mathcal {D}_{\mathfrak {p}}=(A^{\prime }/\mathcal {D})[p]_{\mathfrak {p}}$
. Otherwise, Lemma 3.8 yields
$\mathrm {Hdg}_{\beta }(A^{\prime })=p/(p+1)$
for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
. By [Reference Hattori29, Proposition 6.1], we see that
$$ \begin{align*} \deg_{\beta}\left(A^{\prime}[p]_{\mathfrak{p}}/\mathcal{D}_{\mathfrak{p}}\right)=p/(p+1),\qquad \mathrm{Hdg}_{\beta}\left((A^{\prime}/\mathcal{D})[p]_{\mathfrak{p}}\right)=1/(p+1) \end{align*} $$
for any
$\beta \in \mathbb {B}_{\mathfrak {p}}$
and that
$(A^{\prime }/\mathcal {D})[p]_{\mathfrak {p}}$
has the canonical subgroup
$A^{\prime }[p]_{\mathfrak {p}}/\mathcal {D}_{\mathfrak {p}}$
. Hence the HBAV
$A^{\prime }/\mathcal {D}$
satisfies
$$ \begin{align*} \mathrm{Hdg}_{\beta}(A^{\prime}/\mathcal{D})\leq 1/(p+1) \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
and has the canonical subgroup
$$ \begin{align*} A^{\prime}[p]/\mathcal{D}=\bigoplus_{\mathfrak{p}\mid p} A^{\prime}[p]_{\mathfrak{p}}/\mathcal{D}_{\mathfrak{p}} \end{align*} $$
of level
$1$
. This concludes the proof of the corollary.
Lemma 3.10. Suppose
$f_{\mathfrak {p}}\leq 2$
for any
$\mathfrak {p}\mid p$
. Then we have
$$ \begin{align*} \mathcal{V}_{\mathfrak{c},Q}\left(\tfrac{1}{p+1}\right)\neq \emptyset. \end{align*} $$
Moreover, for any finite extension
$L/K$
and any element
$[(A^{\prime },\mathcal {H}^{\prime })]$
of
$Y_{\mathfrak {c},p}(\mathcal {O}_{L})$
satisfying
$\left [\left (A^{\prime }_{L},\mathcal {H}^{\prime }_{L}\right )\right ]\in \mathcal {V}_{\mathfrak {c},Q}\left (\tfrac {1}{p+1}\right )(L)$
, we have
$\mathrm {Hdg}_{\beta }(A^{\prime })\leq 1/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
and the HBAV
$A^{\prime }$
has the canonical subgroup
$\mathcal {H}^{\prime }$
.
Proof. Recall that we have
$\mathrm {sp}^{-1}\left (\bar {Q}\right )=(\mathrm {Spf}(\mathfrak {B}^{\prime }))^{\mathrm {rig}}$
with the ring
$\mathfrak {B}^{\prime }$
of equation (3.4) in the proof of Proposition 3.7. From the description of
$\deg _{\beta }$
in terms of the parameter
$X_{\beta }$
of the ring
$\mathfrak {B}^{\prime }$
, we see that there exists a point
$[(A^{\prime },\mathcal {H}^{\prime })]\in Y_{\mathfrak {c},p}(\mathcal {O}_{L})$
with some finite extension
$L/K$
such that
$\left [\left (A^{\prime }_{L},\mathcal {H}^{\prime }_{L}\right )\right ]\in \mathrm {sp}^{-1}\left (\bar {Q}\right )$
and
$$ \begin{align*} \deg_{\beta}(A^{\prime}[p]/\mathcal{H}^{\prime})=1/(p+1) \end{align*} $$
for any
$\beta \in \mathbb {B}_{F}$
. Then [Reference Hattori29, Lemma 5.1(1)] implies that
$\mathrm {Hdg}_{\beta }(A^{\prime })=1/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
and thus
$\left [\left (A^{\prime }_{L},\mathcal {H}^{\prime }_{L}\right )\right ]\in \mathcal {V}_{\mathfrak {c},Q}\left (\tfrac {1}{p+1}\right )(L)$
. The last assertion also follows from [Reference Hattori29, Lemma 5.1(1)].
Since
$\mathcal {Y}_{\mathfrak {c},p}$
is separated, Proposition 3.7 implies that the base extension
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}=\mathcal {V}_{\mathfrak {c},Q}\hat {\otimes }_{K} \mathbb {C}_{p}$
is an admissible affinoid open subset of
$\mathcal {Y}_{\mathfrak {c},p,\mathbb {C}_{p}}$
whose connected components are all isomorphic to
$\mathcal {V}_{\mathfrak {c},Q}\hat {\otimes }_{K^{\prime }} \mathbb {C}_{p}$
. Each connected component contains an affinoid subdomain of
$\mathcal {V}_{\mathfrak {c},Q}\left (\tfrac {1}{p+1}\right )\hat {\otimes }_{K} \mathbb {C}_{p}$
which is isomorphic to
$\mathcal {V}_{\mathfrak {c},Q}\left (\tfrac {1}{p+1}\right ) \hat {\otimes }_{K^{\prime }} \mathbb {C}_{p}$
. By Lemma 3.10, we have
$$ \begin{align*} \mathcal{V}_{\mathfrak{c},Q}\left(\tfrac{1}{p+1}\right)\hat{\otimes}_{K^{\prime}} \mathbb{C}_{p}\neq \emptyset. \end{align*} $$
The point
$Q\in \mathcal {Y}_{\mathfrak {c},p}(L)$
defines a point of
$\mathcal {Y}_{\mathfrak {c},p,\mathbb {C}_{p}}\left (\mathbb {C}_{p}\right )$
by the natural inclusion
$L\to \mathbb {C}_{p}$
, which we also denote by Q. Let
$ \mathcal {V}^{0}_{\mathfrak {c},Q,\mathbb {C}_{p}}$
be the connected component of
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}$
containing Q and let
$\mathcal {V}^{0}_{\mathfrak {c},Q,\mathbb {C}_{p}}\left (\tfrac {1}{p+1}\right )$
be a copy of
$\mathcal {V}_{\mathfrak {c},Q}\left (\tfrac {1}{p+1}\right ) \hat {\otimes }_{K^{\prime }} \mathbb {C}_{p}$
which is contained in
$\mathcal {V}^{0}_{\mathfrak {c},Q,\mathbb {C}_{p}}$
. These are both nonempty admissible affinoid open subsets of
$\mathcal {Y}_{\mathfrak {c},p,\mathbb {C}_{p}}$
.
3.3 Overconvergent Hilbert modular forms and the eigenvariety
In this subsection, we recall the construction of sheaves of overconvergent Hilbert modular forms and the associated eigenvariety, due to Andreatta, Iovita, and Pilloni [Reference Andreatta, Iovita and Pilloni4].
3.3.1 Overconvergent modular forms over Hilbert modular varieties
Put
$\mathbb {T}=\mathrm {Res}_{\mathcal {O}_{F}/\mathbb {Z}}(\mathbb {G}_{\mathrm {m}})$
. Let
$\hat {\mathbb {T}}$
be its formal completion along the unit section. For any
$w \in e^{-1}\mathbb {Z}_{\geq 1}$
, let
$\mathbb {T}_{w}^{0}$
be the formal subgroup scheme of
$\hat {\mathbb {T}}$
over
$\mathrm {Spf}(\mathcal {O}_{K})$
representing the functor
$$ \begin{align*} \mathfrak{B} \mapsto \mathrm{Ker}\left(\mathbb{T}(\mathfrak{B})\to \mathbb{T}\left(\mathfrak{B}/\pi_{K}^{e w}\mathfrak{B}\right)\right) \end{align*} $$
on the category of admissible formal
$\mathcal {O}_{K}$
-algebras
$\mathfrak {B}$
. Then
$\mathbb {T}^{0}_{w}$
is a quasi-compact admissible formal group scheme over
$\mathcal {O}_{K}$
.
Let
$\mathcal {W}$
be the Berthelot generic fiber of
$\mathrm {Spf}\left (\mathcal {O}_{K}\left [\left [\mathbb {T}\left (\mathbb {Z}_{p}\right )\right ]\right ]\right )$
and denote the universal character on this space by
$$ \begin{align*} \kappa^{\mathrm{un}}: \mathbb{T}\left(\mathbb{Z}_{p}\right) \to \mathcal{O}^{\circ}(\mathcal{W})^{\times}=\mathcal{O}_{K}\left[\left[\mathbb{T}\left(\mathbb{Z}_{p}\right)\right]\right]^{\times}. \end{align*} $$
Here
$\mathcal {O}^{\circ }$
is the sheaf of rigid analytic functions with absolute value bounded by
$1$
and the last equality follows from [Reference de Jong19, Theorem 7.4.1]. For any morphism
$\mathcal {X}\to \mathcal {W}$
of rigid analytic varieties over K, we denote by
$\kappa ^{\mathcal {X}}$
the restriction
$$ \begin{align*} \kappa^{\mathcal{X}}:\mathbb{T}\left(\mathbb{Z}_{p}\right)\overset{\kappa^{\mathrm{un}}}{\to}\mathcal{O}^{\circ}(\mathcal{W})^{\times} \to \mathcal{O}^{\circ}(\mathcal{X})^{\times} \end{align*} $$
of
$\kappa ^{\mathrm {un}}$
to
$\mathcal {X}$
. Consider the case where
$\mathcal {X}$
is a reduced K-affinoid variety
$\mathcal {U}=\mathrm {Sp}(A)$
. Then the subring
$A^{\circ }$
of power-bounded elements is p-adically complete. For any positive integer n, put
$q_{n}=2$
if
$p=2$
and
$n=1$
, and
$q_{n}=1$
otherwise. When we consider the case of
$p=2$
and
$n=1$
, we assume that
$2$
splits completely in F. The character
$\kappa ^{\mathcal {U}}$
is said to be n-analytic if the restriction to
$\mathbb {T}^{0}_{n}\left (\mathbb {Z}_{p}\right )$
factors as
with some
$\mathbb {Z}_{p}$
-linear map
$\psi $
. In this case, we also say that the morphism
$\mathcal {U}\to \mathcal {W}$
is n-analytic. Any
$\kappa ^{\mathcal {U}}$
is n-analytic for some n by the maximal modulus principle. Note that any n-analytic character defines an analytic character
$\mathbb {T}^{0}_{n}\left (\mathbb {Z}_{p}\right )\to A^{\times }$
, even for the case of
$p=2$
and
$n=1$
. Moreover, put
$$ \begin{align*} \delta_{n}= \begin{cases} \frac{1}{p-1}& (p\geq 3)\\ \frac{1}{4}& (p=2,\ n=1)\\ 0& (p=2,\ n\geq 2). \end{cases} \end{align*} $$
Then for any
$w\in e^{-1}\mathbb {Z}$
satisfying
$$ \begin{align*} n-1+\delta_{n}<w\leq n, \end{align*} $$
any n-analytic character extends to an analytic character
$\mathbb {T}^{0}_{w}(\mathcal {O}_{K})\to A^{\times }$
.
Using [Reference Hattori29, Theorem 8.1] and Lemma 3.5, we can generalize the construction in [Reference Andreatta, Iovita and Pilloni4, §3.3]. Let n be a positive integer and put
$$ \begin{align} B_{n}= \begin{cases} \frac{p-2}{p^{n}}& (p\geq 3)\\ \frac{3}{8}& (p=2,\ n=1)\\ \frac{p-1}{p^{n}}& (p=2,\ n\geq 2), \end{cases} \end{align} $$
so that we have
$$ \begin{align*} \frac{1}{p+1}<B_{1}. \end{align*} $$
Let
$\underline {v}=\left (v_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
be a g-tuple in
$[0,B_{n})\cap \mathbb {Q}$
. Put
$v=\max \left \{v_{\beta }\mid \beta \in \mathbb {B}_{F}\right \}$
. Let
$C_{n}$
be the canonical subgroup of
$\bar {\mathcal {A}}^{\mathrm {un}}$
of level n over
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
, as before. Put
$$ \begin{align*} \bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})\left(\underline{v}\right)=\mathrm{Isom}_{\bar{\mathcal{M}}\left(\mu_{N},\mathfrak{c}\right)\left(\underline{v}\right)}\left(C_{n}, \mathcal{D}_{F}^{-1}\otimes \mu_{p^{n}}\right). \end{align*} $$
We denote by
$\bar {\mathfrak {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
the normalization of
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
in
$\bar {\mathcal {M}}(\Gamma _{1}(p^{n}), \mu _{N}, \mathfrak {c})\left (\underline {v}\right )$
. Note that since
$\mathcal {C}^{\vee }_{n}$
is finite and étale over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
, we have
$$ \begin{align} \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(0)=\mathrm{Isom}_{\bar{\mathfrak{M}}\left(\mu_{N},\mathfrak{c}\right)(0)}\left(\mathcal{C}_{n}, \mathcal{D}_{F}^{-1}\otimes \mu_{p^{n}}\right), \end{align} $$
which is a
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
-torsor over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
.
Let w be an element of
$e^{-1}\mathbb {Z}$
satisfying
$$ \begin{align*} n-1+\delta_{n}< w\leq n-\frac{v(p^{n}-1)}{p-1}-\delta_{K}(v), \end{align*} $$
which exists if
$$ \begin{align} 1-\delta_{n}-\frac{p^{n} v}{p-1}\geq \frac{2}{e}. \end{align} $$
Note that condition (3.7) is satisfied for a sufficiently large K. Let
$\mathcal {F}$
be the locally free
$\mathcal {O}_{F}\otimes \mathcal {O}_{\bar {\mathfrak {M}}\left (\Gamma _{1}\left (p^{n}\right ),\mu _{N},\mathfrak {c}\right )\left (\underline {v}\right )}$
-module of rank
$1$
constructed as in [Reference Andreatta, Iovita and Pilloni4, Proposition 3.4], using Lemma 3.5 for
$i=w+\delta _{K}(v)$
. Let
$$ \begin{align*} \gamma_{w}: \mathfrak{IW}_{w}^{+} \to \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}),\mu_{N}, \mathfrak{c})\left(\underline{v}\right) \end{align*} $$
be the p-adic formal
$\mathbb {T}^{0}_{w}$
-torsor over
$\bar {\mathfrak {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
classifying, for any
$R\in \mathbf {NAdm}$
and any morphism of p-adic formal schemes
$\gamma :\mathrm {Spf}(R)\to \bar {\mathfrak {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
, the isomorphisms
$\alpha : \gamma ^{*}\mathcal {F}\to \mathcal {O}_{F}\otimes R$
such that the composite
$$ \begin{align*} \underline{\mathcal{O}_{F}/p^{n} \mathcal{O}_{F}}(R) \overset{\gamma}{\simeq} \mathcal{C}_{n}^{\vee}(R)\overset{\mathrm{HT}_{w}}{\to} \gamma^{*}\mathcal{F}/\pi_{K}^{ew} \gamma^{*}\mathcal{F} \overset{\alpha}{\simeq} \mathcal{O}_{F}\otimes R/\pi_{K}^{e w} R \end{align*} $$
sends
$1$
to
$1$
[Reference Andreatta, Iovita and Pilloni4, §3.4]. We also write
$\mathfrak {IW}_{w}^{+}$
as
$\mathfrak {IW}^{+}_{w,\mathfrak {c}}\left (\underline {v}\right )$
. We denote the Raynaud generic fiber of
$\mathfrak {IW}_{w}^{+}$
by
$\mathcal {IW}_{w}^{+}$
and also by
$\mathcal {IW}^{+}_{w,\mathfrak {c}}\left (\underline {v}\right )$
. From equality (3.6), we see that the moduli interpretation of
$\mathfrak {IW}_{w,\mathfrak {c}}^{+}(0)$
already given is also valid for the category of quasi-idyllic p-adic
$ \mathcal {O}_{K}$
-algebras R.
For the structure morphism
$$ \begin{align*} h_{n}: \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})\left(\underline{v}\right)\to \bar{\mathfrak{M}}(\mu_{N},\mathfrak{c})\left(\underline{v}\right), \end{align*} $$
we put
$\pi _{w}=h_{n}\circ \gamma _{w}$
. We denote by
$\gamma _{w}^{\mathrm {rig}}$
,
$h_{n}^{\mathrm {rig}}$
, and
$\pi _{w}^{\mathrm {rig}}$
the induced morphisms on the Raynaud generic fibers. Let
$\mathbb {T}_{w}$
be the formal subgroup scheme of
$\hat {\mathbb {T}}$
over
$\mathrm {Spf}(\mathcal {O}_{K})$
whose set of
$\mathfrak {B}$
-valued points are the inverse image of
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
by the map
$\mathbb {T}(\mathfrak {B})\to \mathbb {T}\left (\mathfrak {B}/\pi _{K}^{e w}\mathfrak {B}\right )$
for any admissible formal
$\mathcal {O}_{K}$
-algebra
$\mathfrak {B}$
. The natural action of
$\mathbb {T}_{w}^{0}$
on
$\mathfrak {IW}_{w}^{+}$
induces an action of
$\mathbb {T}_{w}$
on
$\mathfrak {IW}_{w}^{+}$
over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
and also on the Raynaud generic fiber
$\mathcal {IW}_{w}^{+}$
over
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
. Then for any reduced K-affinoid variety
$\mathcal {U}$
and n-analytic morphism
$\mathcal {U}\to \mathcal {W}$
, we define
$$ \begin{align*} \Omega^{\kappa^{\mathcal{U}}}=\left(\pi_{w}^{\mathrm{rig}}\right)_{*}\left(\mathcal{O}_{\mathcal{IW}_{w}^{+}\times \mathcal{U}}\right)\left[-\kappa^{\mathcal{U}}\right]. \end{align*} $$
By [Reference Andreatta, Iovita and Pilloni4, Proposition 3.13], it is an invertible sheaf which is independent of the choices of n and w. Let D be the boundary divisor of
$\bar {\mathcal {M}} (\mu _{N},\mathfrak {c})$
. We also put
$$ \begin{align*} M\left(\mu_{N}, \mathfrak{c}, \kappa^{\mathcal{U}}\right)\left(\underline{v}\right)&=H^{0}\left(\bar{\mathcal{M}}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)\times \mathcal{U}, \Omega^{\kappa^{\mathcal{U}}}\right),\\ S\left(\mu_{N},\mathfrak{c}, \kappa^{\mathcal{U}}\right)\left(\underline{v}\right)&=H^{0}\left(\bar{\mathcal{M}}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)\times \mathcal{U}, \Omega^{\kappa^{\mathcal{U}}}(-D)\right). \end{align*} $$
For any
$R\in \mathbf {NAdm}$
, let us consider tuples
$(A, \iota ,\lambda ,\psi ,u,\alpha )$
over R consisting of an HBAV
$(A,\iota , \lambda ,\psi )$
over
$\mathrm {Spec}(R)$
such that
$\mathrm {Hdg}_{\beta }(A_{x})\leq v_{\beta }$
for any
$x\in \mathrm {Sp}(R[1/p])$
, an isomorphism of
$\mathcal {O}_{F}$
-group schemes
$$ \begin{align*} u: \mathcal{C}_{n}\rvert_{R\left[1/p\right]}\simeq \mathcal{D}_{F}^{-1}\otimes\mu_{p^{n}} \end{align*} $$
for the canonical subgroup
$\mathcal {C}_{n}$
of A, and an isomorphism
$$ \begin{align*} \alpha:\gamma^{*}\mathcal{F}\simeq \mathcal{O}_{F}\otimes R \end{align*} $$
satisfying the compatibility with u as before. By equality (3.1), any element
$f\in H^{0}\left (\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )^{\mathrm {rig}}, \Omega ^{\kappa ^{\mathcal {U}}}\right )$
can be identified with a rule functorially associating, with any such tuple over R endowed with a map
$\mathrm {Sp}(R[1/p])\to \mathcal {U}$
, an element
$f(A,\iota ,\lambda ,\psi ,u,\alpha )$
of
$R[1/p]$
satisfying
$$ \begin{align*} f\left(A,\iota,\lambda,\psi,t^{-1}u,t^{-1}\alpha\right)=\kappa^{\mathcal{U}}(t) f(A,\iota,\lambda,\psi,u,\alpha) \end{align*} $$
for any
$t\in \mathbb {T}\left (\mathbb {Z}_{p}\right )$
. Similarly, any element
$f\in H^{0}\left (\mathfrak {M}(\mu _{N},\mathfrak {c})(0)^{\mathrm {rig}}, \Omega ^{\kappa ^{\mathcal {U}}}\right )$
has a similar description as a rule over any idyllic p-adic
$\mathcal {O}_{K}$
-algebra R endowed with a morphism
$\mathrm {Spf}(R)\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
.
For later use, we also recall the definition of an integral structure of the sheaf
$\Omega ^{\kappa ^{\mathcal {U}}}$
for an n-analytic map
$\kappa ^{\mathcal {U}}: \mathcal {U}=\mathrm {Sp}(A)\to \mathcal {W}$
with some reduced K-affinoid algebra A. Note that
$A^{\circ }$
is topologically of finite type [Reference Bosch, Güntzer and Remmert10, Corollary 6.4.1/6] and thus
$\mathfrak {U}=\mathrm {Spf}(A^{\circ })$
is an admissible formal scheme over
$\mathrm {Spf}(\mathcal {O}_{K})$
. The map
$\kappa ^{\mathcal {U}}$
extends to a formal character
$$ \begin{align*} \boldsymbol{\kappa}^{\mathcal{U}}: \mathbb{T}_{w}\times \mathfrak{U}\to \hat{\mathbb{G}}_{\mathrm{m}}\times \mathfrak{U}. \end{align*} $$
We put
$$ \begin{align*} \mathbf{\Omega}^{\kappa^{\mathcal{U}}}=(\pi_{w})_{*}\left(\mathcal{O}_{\mathfrak{IW}_{w}^{+}\times \mathfrak{U}}\right)\left[-\boldsymbol{\kappa}^{\mathcal{U}}\right]. \end{align*} $$
It is a coherent
$\mathcal {O}_{\bar {\mathfrak {M}}\left (\mu _{N},\mathfrak {c}\right )\left (\underline {v}\right )\times \mathfrak {U}}$
-module which is independent of the choice of w such that its Raynaud generic fiber is
$\Omega ^{\kappa ^{\mathcal {U}}}$
[Reference Andreatta, Iovita and Pilloni4, Proposition 3.13]. Since the map
$h_{n}$
is an étale
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
-torsor over the ordinary locus
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0) $
, the restriction of
$\mathbf {\Omega }^{\kappa ^{\mathcal {U}}}$
to
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)\times \mathfrak {U}$
is an invertible sheaf.
Let
$\kappa :\mathbb {T}\left (\mathbb {Z}_{p}\right )\to K^{\times }$
be a weight character which is integral – namely, it is written as
$$ \begin{align*} \mathbb{T}\left(\mathbb{Z}_{p}\right)=\left(\mathcal{O}_{F}\otimes \mathbb{Z}_{p}\right)^{\times} \ni t\otimes 1\mapsto \prod_{\beta\in \mathbb{B}_{F}}\beta(t)^{k_{\beta}} \in K^{\times} \end{align*} $$
with some g-tuple of integers
$\left (k_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
. In this case, the sheaf
$\Omega ^{\kappa }$
is isomorphic to the classical automorphic sheaf [Reference Andreatta, Iovita and Pilloni4, Corollary 3.10]. Indeed, consider
$\mathcal {I}=\mathrm {Isom}_{\bar {\mathcal {M}}\left (\mu _{N},\mathfrak {c}\right )}\left (\mathcal {O}_{F}\otimes \mathcal {O}_{\bar {\mathcal {M}}\left (\mu _{N},\mathfrak {c}\right )}, \omega _{\bar {\mathcal {A}}^{\mathrm {un}}}\right )$
. Since the Raynaud generic fiber of the sheaf
$\mathcal {F}$
is
$\omega _{\bar {\mathcal {A}}^{\mathrm {un}}}$
, we have a natural map
$\mathcal {IW}_{w}^{+}\to \mathcal {I}$
, which induces an isomorphism
$\omega _{\bar {\mathcal {A}} {}^{\mathrm {un}}}^{\kappa }\to \Omega ^{\kappa }$
. We also say that an integral weight
$\kappa $
is doubly even if every
$k_{\beta }$
is divisible by
$4$
.
Moreover, we say that a weight character
$\kappa :\mathbb {T}\left (\mathbb {Z}_{p}\right )\to K^{\times }$
is n-integral (resp., n-doubly even) if its restriction to
$\mathbb {T}^{0}_{n}\left (\mathbb {Z}_{p}\right )$
is equal to the restriction of a character of some integral (resp., doubly even) weight
$\left (k_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
. Then, from the construction of the sheaf
$\Omega ^{\kappa }$
, we see that the pullback
$\left (h^{\mathrm {rig}}_{n}\right )^{*}\Omega ^{\kappa }$
to
$\bar {\mathcal {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
is isomorphic to
$\left (h^{\mathrm {rig}}_{n}\right )^{*} \left (\bigotimes _{\beta \in \mathbb {B}_{F}} \omega ^{\otimes k_{\beta }}_{\bar {\mathcal {A}}^{\mathrm {un}},\beta }\right )$
. Note that for the case where
$p=2$
splits completely in F, a
$1$
-integral weight is
$1$
-analytic if and only if it is
$1$
-doubly even.
3.3.2 Overconvergent arithmetic Hilbert modular forms
We define the weight space
$\mathcal {W}^{G}$
for overconvergent Hilbert modular forms as the Berthelot generic fiber of
$\mathrm {Spf}\left (\mathcal {O}_{K}\left [\left [\mathbb {T}\left (\mathbb {Z}_{p}\right )\times \mathbb {Z}_{p}^{\times }\right ]\right ]\right )$
. Any morphism
$\mathcal {X}\to \mathcal {W}^{G}$
defines a pair
$\left (\nu ^{\mathcal {X}}, w^{\mathcal {X}}\right )$
of continuous characters
$$ \begin{align*} \nu^{\mathcal{X}}: \mathbb{T}\left(\mathbb{Z}_{p}\right)\to \mathcal{O}^{\circ}(\mathcal{X})^{\times},\qquad w^{\mathcal{X}}:\mathbb{Z}_{p}^{\times} \to \mathcal{O}^{\circ}(\mathcal{X})^{\times} \end{align*} $$
with respect to the supremum seminorm on
$\mathcal {X}$
. The map
$$ \begin{align*} \mathbb{T}\left(\mathbb{Z}_{p}\right)\to \mathbb{T}\left(\mathbb{Z}_{p}\right)\times \mathbb{Z}_{p}^{\times},\quad t\mapsto \left(t^{2}, \mathrm{N}_{F/\mathbb{Q}}(t)\right), \end{align*} $$
induces a morphism
$k: \mathcal {W}^{G}\to \mathcal {W}$
. For any morphism
$\mathcal {X}\to \mathcal {W}^{G}$
, put
$\kappa ^{\mathcal {X}}=k\left (\nu ^{\mathcal {X}},w^{\mathcal {X}}\right )$
. When
$\mathcal {X}$
is a reduced K-affinoid variety, we say that
$\left (\nu ^{\mathcal {X}},w^{\mathcal {X}}\right )$
is n-analytic if
$\nu ^{\mathcal {X}}$
and
$w^{\mathcal {X}}$
are both n-analytic. Note that if
$\left (\nu ^{\mathcal {X}},w^{\mathcal {X}}\right )$
is n-analytic, then
$\kappa ^{\mathcal {X}}$
is also n-analytic. We say that a character
$(\nu ,w): \mathbb {T}\left (\mathbb {Z}_{p}\right )\times \mathbb {Z}_{p}^{\times }\to K^{\times }$
is integral if it comes from an algebraic character
$\mathbb {T}\times \mathbb {G}_{\mathrm {m}}\to \mathbb {G}_{\mathrm {m}}$
. Then it is written as
$$ \begin{align*} \mathbb{T}\left(\mathbb{Z}_{p}\right)\times \mathbb{Z}_{p}^{\times}\to K^{\times},\quad (t\otimes 1,s)\mapsto \prod_{\beta\in \mathbb{B}_{F}} \beta(t)^{k_{\beta}}s^{k_{0}}, \end{align*} $$
with some g-tuple of integers
$\left (k_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
and an integer
$k_{0}$
. We say that it is doubly even if every
$k_{\beta }$
and
$k_{0}$
are divisible by
$4$
. We also say that
$(\nu ,w)$
is n-integral (resp., n-doubly even) if its restriction to
$\mathbb {T}^{0}_{n}\left (\mathbb {Z}_{p}\right )\times \left (1+p^{n}\mathbb {Z}_{p}\right )$
is equal to the restriction of some integral (resp., doubly even) character. If
$(\nu ,w)$
is n-integral (resp., n-doubly even), then
$k(\nu ,w)$
is also n-integral (resp., n-doubly even).
Let
$\mathcal {U}$
be a reduced K-affinoid variety and let
$\mathcal {U}\to \mathcal {W}^{G}$
be an n-analytic morphism. Note that for any
$\mathfrak {c}$
-polarization
$\lambda : A\otimes _{\mathcal {O}_{F}} \mathfrak {c}\to A^{\vee }$
and any
$x\in F^{\times ,+}$
, multiplication by x gives an
$x^{-1}\mathfrak {c}$
-polarization
$$ \begin{align*} x\lambda: A\otimes_{\mathcal{O}_{F}} x^{-1}\mathfrak{c}\overset{\times x}{\simeq} A\otimes_{\mathcal{O}_{F}} \mathfrak{c}\overset{\lambda}{\simeq} A^{\vee}. \end{align*} $$
Then the group
$\Delta =\mathcal {O}_{F}^{\times ,+}/U_{N}^{2}$
acts on the space
$M\left (\mu _{N}, \mathfrak {c}, \kappa ^{\mathcal {U}}\right )\left (\underline {v}\right )$
by
$$ \begin{align*} ([\epsilon].f)(A,\iota,\lambda,\psi, u,\alpha)=\nu^{\mathcal{U}}(\epsilon)f\left(A,\iota,\epsilon^{-1}\lambda,\psi, u,\alpha\right) \end{align*} $$
for any
$f\in M\left (\mu _{N}, \mathfrak {c}, \kappa ^{\mathcal {U}}\right )\left (\underline {v}\right )$
and
$\epsilon \in \mathcal {O}_{F}^{\times ,+}$
. We define
$$ \begin{align*} M^{G}\left(\mu_{N},\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right)&=M\left(\mu_{N}, \mathfrak{c}, \kappa^{\mathcal{U}}\right)\left(\underline{v}\right)^{\Delta},\\ S^{G}\left(\mu_{N},\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right)&=S\left(\mu_{N}, \mathfrak{c}, \kappa^{\mathcal{U}}\right)\left(\underline{v}\right)^{\Delta}. \end{align*} $$
Let
$F^{\times , +,(p)}$
be the subgroup of
$F^{\times ,+}$
consisting of p-adic units. For any
$x\in F^{\times ,+,(p)}$
, we define a map
$$ \begin{align*} L_{x}: M^{G}\left(\mu_{N},\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right) \to M^{G}\left(\mu_{N},x^{-1}\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right) \end{align*} $$
by the formula
$$ \begin{align*} (L_{x}(f))(A,\iota,\lambda,\psi,u,\alpha)=\nu^{\mathcal{U}}(x) f\left(A,\iota, x^{-1}\lambda,\psi,u,\alpha\right). \end{align*} $$
Let
$\mathrm {Frac}(F)^{(p)}$
be the group of fractional ideals of F which are prime to p. Then the spaces
$$ \begin{align*} M^{G}\left(\mu_{N}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right),\qquad S^{G}\left(\mu_{N}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right) \end{align*} $$
of arithmetic overconvergent Hilbert modular forms and cusp forms are defined as the quotients
$$ \begin{align*} &\left(\bigoplus_{\mathfrak{c}\in \mathrm{Frac}(F)^{\left(p\right)}} M^{G}\left(\mu_{N},\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right)\right)/\left(L_{x}(f)-f\mid x\in F^{\times, +,(p)}\right),\\ &\left(\bigoplus_{\mathfrak{c}\in \mathrm{Frac}(F)^{\left(p\right)}} S^{G}\left(\mu_{N},\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(\underline{v}\right)\right)/\left(L_{x}(f)-f\mid x\in F^{\times, +,(p)}\right). \end{align*} $$
By the same construction, we also have the spaces
$$ \begin{align*} M^{G}\left(\mu_{N}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}}),\qquad S^{G}\left(\mu_{N}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}}). \end{align*} $$
3.3.3 Hecke operators and the Hilbert eigenvariety
Next we recall the definition of Hecke operators on the space of overconvergent Hilbert modular forms, following [Reference Andreatta, Iovita and Pilloni4, §3.7]. Let n,
$\underline {v}$
, v, and w be as before. For any HBAV
$(A,\iota ,\lambda ,\psi )$
over a base scheme
$S/\mathrm {Spec}(\mathcal {O}_{K})$
, the closed immersion
$\psi : \mathcal {D}^{-1}_{F}\otimes \mu _{N} \to A$
gives a subgroup scheme
$\mathrm {Im}(\psi )$
of A which is étale locally isomorphic to
$\underline {\mathcal {D}_{F}^{-1}/N\mathcal {D}_{F}^{-1}}$
. Let
$\mathfrak {l}$
be any nonzero ideal of
$\mathcal {O}_{F}$
. We define
$$ \begin{align*} \mathcal{Y}^{\prime}_{\mathfrak{c},\mathfrak{l}}\left(\underline{v}\right)\subseteq \mathcal{M}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)\times \mathcal{M}(\mu_{N},\mathfrak{l}\mathfrak{c})\left(\underline{v}\right) \end{align*} $$
as the subvariety classifying pairs
$((A,\iota ,\lambda ,\psi ),(A^{\prime },\iota ^{\prime },\lambda ^{\prime },\psi ^{\prime }))$
and an isogeny
$\pi _{\mathfrak {l}}:A\to A^{\prime }$
compatible with the other data such that
$\mathrm {Ker}(\pi _{\mathfrak {l}})$
is étale locally isomorphic to
$\underline {\mathcal {O}_{F}/\mathfrak {l} \mathcal {O}_{F}}$
,
$\mathrm {Ker}(\pi _{\mathfrak {l}})\cap \mathrm {Im}(\psi )=0$
, and
$\mathrm {Ker}(\pi _{\mathfrak {l}})\cap C_{1}=0$
, where
$C_{1}$
is the canonical subgroup of A of level
$1$
. Consider the projections
$$ \begin{align*} p_{1}: \mathcal{Y}^{\prime}_{\mathfrak{c},\mathfrak{l}}\left(\underline{v}\right)\to \mathcal{M}(\mu_{N},\mathfrak{c})\left(\underline{v}\right),\qquad p_{2}: \mathcal{Y}^{\prime}_{\mathfrak{c},\mathfrak{l}}\left(\underline{v}\right)\to \mathcal{M}(\mu_{N},\mathfrak{l}\mathfrak{c})\left(\underline{v}\right). \end{align*} $$
Note that map
$p_{1}$
is finite and étale. For the case where
$\mathfrak {l}$
is a prime ideal dividing p, we suppose that
$p^{-1}v_{\sigma \circ \beta }\leq v_{\beta }$
for any
$\beta \in \mathbb {B}_{\mathfrak {l}}$
. Set
$\underline {v}^{\prime }=\left (v^{\prime }_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
by
$v^{\prime }_{\beta }=v_{\beta }$
for
$\beta \notin \mathbb {B}_{\mathfrak {l}}$
and
$v^{\prime }_{\beta }=p^{-1}v_{\sigma \circ \beta }$
for
$\beta \in \mathbb {B}_{\mathfrak {l}}$
. Then [Reference Hattori29, Corollary 5.3(1)] implies that map
$p_{2}$
factors through the admissible open subset
$\mathcal {M}(\mu _{N},\mathfrak {l}\mathfrak {c})\left (\underline {v}^{\prime }\right )\subseteq \mathcal {M}(\mu _{N},\mathfrak {l}\mathfrak {c})\left (\underline {v}\right )$
.
Let
$\mathcal {U}$
be a smooth K-affinoid variety and let
$\mathcal {U}\to \mathcal {W}$
be an n-analytic map. Then [Reference Hattori29, Theorem 8.1(10)] and the proof of [Reference Andreatta, Iovita and Pilloni4, Corollary 3.26] (see also [Reference Andreatta, Iovita and Pilloni3, Lemma 6.1.1]) show that the map
$\pi _{\mathfrak {l}}^{*}: \omega _{A^{\prime }}\to \omega _{A}$
induces an isomorphism
$$ \begin{align*} \pi_{\mathfrak{l}}: p_{2}^{*} \mathcal{IW}_{w,\mathfrak{l} \mathfrak{c}}^{+}\left(\underline{v}\right) \simeq p_{1}^{*}\mathcal{IW}_{w,\mathfrak{c}}^{+}\left(\underline{v}\right), \end{align*} $$
which in turn defines an isomorphism
$$ \begin{align*} \pi_{\mathfrak{l}}^{*}: p_{1}^{*}\left(\Omega^{\kappa^{\mathcal{U}}}\right)\simeq p_{2}^{*}\left(\Omega^{\kappa^{\mathcal{U}}}\right). \end{align*} $$
This gives the Hecke operator
$$ \begin{align*} H^{0}\left(\mathcal{M}(\mu_{N},\mathfrak{l}\mathfrak{c})\left(\underline{v}\right)\times \mathcal{U},\Omega^{\kappa^{\mathcal{U}}}\right) & \overset{p_{2}^{*}}{\to} H^{0}\left(\mathcal{Y}^{\prime}_{\mathfrak{c},\mathfrak{l}}\left(\underline{v}\right)\times \mathcal{U},p_{2}^{*}\Omega^{\kappa^{\mathcal{U}}}\right)\\ & \overset{\left(\pi_{\mathfrak{l}}^{*}\right)^{-1}}{\to} H^{0}\left(\mathcal{Y}^{\prime}_{\mathfrak{c}, \mathfrak{l}}\left(\underline{v}\right)\times \mathcal{U},p_{1}^{*}\Omega^{\kappa^{\mathcal{U}}}\right)\\ & \overset{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{l})^{-1}\mathrm{Tr}_{p_{1}}}{\to} H^{0}\left(\mathcal{M}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)\times \mathcal{U},\Omega^{\kappa^{\mathcal{U}}}\right), \end{align*} $$
which can be seen as a map
$M\left (\mu _{N},\mathfrak {l}\mathfrak {c},\kappa ^{\mathcal {U}}\right )\left (\underline {v}\right )\to M\left (\mu _{N},\mathfrak {c},\kappa ^{\mathcal {U}}\right )\left (\underline {v}\right )$
by [Reference Lütkebohmert33, Theorem 1.6]. We denote this map by
$T_{\mathfrak {l}}$
if
$(\mathfrak {l},p)=1$
and
$T^{\prime }_{\mathfrak {l}}$
otherwise.
On the other hand, for any ideal
$\mathfrak {l}$
with
$(\mathfrak {l},pN)=1$
, we have a map
$$ \begin{align*} s_{\mathfrak{l}}: \mathcal{M}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)\to \mathcal{M}\left(\mu_{N}, \mathfrak{l}^{2} \mathfrak{c}\right)\left(\underline{v}\right),\quad (A,\iota,\lambda,\psi)\mapsto \left(A\otimes_{\mathcal{O}_{F}} \mathfrak{l}^{-1}, \iota^{\prime},\mathfrak{l}^{2}\lambda, \psi^{\prime}\right). \end{align*} $$
Here
$\iota ^{\prime }$
and
$\psi ^{\prime }$
are induced by
$\iota $
and
$\psi $
via the natural isogeny
$A\to A/A[\mathfrak {l}]\simeq A\otimes _{\mathcal {O}_{F}} \mathfrak {l}^{-1}$
, and
$\mathfrak {l}^{2}\lambda $
is the
$\mathfrak {l}^{2}\mathfrak {c}$
-polarization on
$A\otimes _{\mathcal {O}_{F}} \mathfrak {l}^{-1}$
defined by
$$ \begin{align*} \left(A\otimes_{\mathcal{O}_{F}} \mathfrak{l}^{-1}\right)\otimes_{\mathcal{O}_{F}} \mathfrak{l}^{2}\mathfrak{c} =\left(A\otimes {}_{\mathcal{O}_{F}}\mathfrak{c}\right)\otimes_{\mathcal{O}_{F}}\mathfrak{l}\overset{\lambda\otimes 1}{\simeq} A^{\vee}\otimes_{\mathcal{O}_{F}}\mathfrak{l}\simeq \left(A\otimes_{\mathcal{O}_{F}}\mathfrak{l}^{-1}\right)^{\vee}. \end{align*} $$
Then we can show that there exists a natural isomorphism
$\pi _{\mathfrak {l}}^{*}: \Omega ^{\kappa ^{\mathcal {U}}} \simeq s_{\mathfrak {l}}^{*} \Omega ^{\kappa ^{\mathcal {U}}}$
as in [Reference Andreatta, Iovita and Pilloni3, Lemma 6.1.1], and we define the operator
$$ \begin{align*} S_{\mathfrak{l}}: M\left(\mu_{N},\mathfrak{l}^{2}\mathfrak{c},\kappa^{\mathcal{U}}\right)\left(\underline{v}\right)\to M\left(\mu_{N},\mathfrak{c},\kappa^{\mathcal{U}}\right)\left(\underline{v}\right) \end{align*} $$
by
$S_{\mathfrak {l}}=\mathrm {N}_{F/\mathbb {Q}}(\mathfrak {l})^{-2}\left (\pi _{\mathfrak {l}}^{*}\right )^{-1}\circ s_{\mathfrak {l}}^{*}$
. Since
$s_{\mathfrak {l}}^{m}=\mathrm {id}$
for some positive integer m, every eigenvalue of the operator
$S_{\mathfrak {l}}$
is p-integral.
To define arithmetic Hecke operators for
$\mathfrak {l}$
with
$(\mathfrak {l},p)\neq 1$
, let
$v_{\mathfrak {p}}$
be the normalized additive valuation for any
$\mathfrak {p}\mid p$
. We fix once and for all elements
$x_{\mathfrak {p}}\in F^{\times ,+}$
such that
$v_{\mathfrak {p}}\left (x_{\mathfrak {p}}\right )=1$
and
$v_{\mathfrak {p}^{\prime }}\left (x_{\mathfrak {p}}\right )=0$
for any
$\mathfrak {p}^{\prime }\neq \mathfrak {p}$
satisfying
$\mathfrak {p}^{\prime }\mid p$
. We define a map
$$ \begin{align*} x_{\mathfrak{p}}^{*}: M\left(\mu_{N}, x_{\mathfrak{p}}^{-1}\mathfrak{c}, \kappa^{\mathcal{U}}\right)\left(\underline{v}\right)\to M\left(\mu_{N}, \mathfrak{c},\kappa^{\mathcal{U}}\right)\left(\underline{v}\right) \end{align*} $$
by
$f\mapsto \left ((A,\iota ,\lambda ,\psi )\mapsto f\left (A,\iota ,x_{\mathfrak {p}}\lambda ,\psi \right )\right )$
. Then we denote the composite
$$ \begin{align*} \prod_{\mathfrak{p}\mid p}\left(x_{\mathfrak{p}}^{*}\right)^{v_{\mathfrak{p}}(\mathfrak{l})}\circ T^{\prime}_{\mathfrak{l}}:M\left(\mu_{N}, \prod_{\mathfrak{p}\mid p}x_{\mathfrak{p}}^{-v_{\mathfrak{p}}(\mathfrak{l})}\mathfrak{l}\mathfrak{c}, \kappa^{\mathcal{U}}\right)\left(\underline{v}\right)\to M\left(\mu_{N},\mathfrak{c},\kappa^{\mathcal{U}}\right)\left(\underline{v}\right) \end{align*} $$
by
$T_{\mathfrak {l}}$
. We also write it as
$U_{\mathfrak {l}}$
if
$\mathfrak {l}$
divides a power of p. Then the operators
$T_{\mathfrak {l}}$
for any
$\mathfrak {l}$
and
$S_{\mathfrak {l}}$
for
$(\mathfrak {l}, pN)=1$
define actions on
$M^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )\left (\underline {v}\right )$
and
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )\left (\underline {v}\right )$
which commute with each other. Note that
$T_{\mathfrak {l}\mathfrak {l}^{\prime }}=T_{\mathfrak {l}} T_{\mathfrak {l}^{\prime }}$
if
$(\mathfrak {l},\mathfrak {l}^{\prime })=1$
and that [Reference Hattori29, Theorem 8.1(10)] implies
$$ \begin{align} T_{\mathfrak{m}} T_{\mathfrak{m}^{s-1}}= \begin{cases} T_{\mathfrak{m}^{s}}+\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})S_{\mathfrak{m}} T_{\mathfrak{m}^{s-2}} & (\mathfrak{m}\nmid Np)\\ T_{\mathfrak{m}^{s}} & (\mathfrak{m} \mid Np) \end{cases} \end{align} $$
for any maximal ideal
$\mathfrak {m}$
.
Let v be an element of
$\mathbb {Q}\cap \left (0,\frac {p-1}{p}\right )$
. Note that the foregoing definitions of Hecke operators are also valid for
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v_{\mathrm {tot}})$
. Then the operator
$U_{p}$
is a compact operator acting on
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v_{\mathrm {tot}})$
which factors as
$$ \begin{align*} S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}})\subseteq S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(p^{-1}v_{\mathrm{tot}}\right)\to S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}}) \end{align*} $$
and, for
$v<(p-1)/p^{2}$
, also as
$$ \begin{align*} S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}})\to S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(p v_{\mathrm{tot}})\subseteq S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}}). \end{align*} $$
Let
$\mathbf {T}$
be the polynomial ring over K with variables
$T_{\mathfrak {l}}$
for any
$\mathfrak {l}$
and
$S_{\mathfrak {l}}$
for
$(\mathfrak {l}, pN)=1$
. Then the ring
$\mathbf {T}$
acts on
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v)$
and
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v_{\mathrm {tot}})$
via the Hecke operators defined.
Now we can construct the eigenvariety from these data, as in [Reference Andreatta, Iovita and Pilloni4, §5]. For any positive integer n, we fix a positive rational number
$v_{n}<B_{n}$
satisfying
$v_{n}\geq v_{n+1}$
for any n, where
$B_{n}$
is defined by equation (3.5). We also assume
$v_{n}\leq \left (p-2+v_{p}(2)\right )/p^{n+1}$
for any
$n\geq 2$
, so that replacing K with its finite extension, condition (3.7) for
$v_{n}$
is satisfied for all n.
For any admissible affinoid open subset
$\mathcal {U}\subseteq \mathcal {W}^{G}$
, we put
$$ \begin{align*} n(\mathcal{U})=\min\left\{n\in \mathbb{Z}_{>0}\mid \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\text{ is }\textit{n}\text{-analytic}\right\}. \end{align*} $$
We define a Banach
$\mathcal {O}(\mathcal {U})$
-module
$M_{\mathcal {U}}$
with
$\mathbf {T}$
-action as
$$ \begin{align*} M_{\mathcal{U}}=S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(v_{n(\mathcal{U}),\mathrm{tot}}\right), \end{align*} $$
on which
$U_{p}$
acts as a compact operator. The proof of [Reference Andreatta, Iovita and Pilloni4, Theorem 4.4] remains valid also for
$p=2$
and implies that the
$\mathcal {O}(\mathcal {U})$
-module
$M_{\mathcal {U}}$
satisfies condition
$(Pr)$
. For admissible affinoid open subsets
$\mathcal {U}_{1}\subseteq \mathcal {U}_{2}$
of
$\mathcal {W}^{G}$
, we have
$n(\mathcal {U}_{1})\leq n(\mathcal {U}_{2})$
and [Reference Andreatta, Iovita and Pilloni4, Proposition 3.14] yields a map
$$ \begin{align*} \alpha_{\mathcal{U}_{1},\mathcal{U}_{2}}: M_{\mathcal{U}_{1}}\to S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}_{1}},w^{\mathcal{U}_{1}}\right)\right)\left(v_{n\left(\mathcal{U}_{2}\right),\mathrm{tot}}\right)\simeq M_{\mathcal{U}_{2}}\hat{\otimes}_{\mathcal{O}\left(\mathcal{U}_{2}\right)}\mathcal{O}(\mathcal{U}_{1}), \end{align*} $$
where the first arrow is the restriction map. Note that for any positive rational numbers
$v,v^{\prime }$
satisfying
$v^{\prime }\leq v < pv^{\prime }<(p-1)/p$
, the restriction map
$$ \begin{align*} S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}})\to S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)\left(v^{\prime}_{\mathrm{tot}}\right) \end{align*} $$
is a primitive link. Thus the map
$\alpha _{\mathcal {U}_{1},\mathcal {U}_{2}}$
is a link satisfying the cocycle condition. Hence, by applying the eigenvariety machine [Reference Buzzard12, Construction 5.7], we obtain the Hilbert eigenvariety
$\mathcal {E}\to \mathcal {W}^{G}$
as in [Reference Andreatta, Iovita and Pilloni4, Theorem 5.1].
3.4 The case over
$\mathbb {C}_{p}$
Since we are ultimately interested in overconvergent Hilbert modular forms over
$\mathbb {C}_{p}$
, we need to give a slight generalization of the construction in [Reference Andreatta, Iovita and Pilloni4] over
$\mathbb {C}_{p}$
. As before, for any quasi-separated rigid analytic variety
$\mathcal {X}$
over K and any coherent
$\mathcal {O}_{\mathcal {X}}$
-module
$\mathcal {F}$
, we denote the base extensions of
$\mathcal {X}$
and
$\mathcal {F}$
to
$\mathbb {C}_{p}$
by
$\mathcal {X}_{\mathbb {C}_{p}}$
and
$\mathcal {F}_{\mathbb {C}_{p}}$
, respectively. Similarly, for any quasi-separated admissible formal scheme
$\mathfrak {X}$
over
$\mathrm {Spf}(\mathcal {O}_{K})$
and any coherent
$\mathcal {O}_{\mathfrak {X}}$
-module
$\mathfrak {F}$
, we denote their pullbacks to
$\mathrm {Spf}\left (\mathcal {O}_{\mathbb {C}_{p}}\right )$
by
$\mathfrak {X}_{\mathcal {O}_{\mathbb {C}_{p}}}$
and
$\mathfrak {F}_{\mathcal {O}_{\mathbb {C}_{p}}}$
, respectively. Then on the Raynaud generic fiber we have
$$ \begin{align*} \left(\mathfrak{X}^{\mathrm{rig}}\right)_{\mathbb{C}_{p}}=\left(\mathfrak{X}_{\mathcal{O}_{\mathbb{C}_{p}}}\right)^{\mathrm{rig}},\qquad \left(\mathfrak{F}^{\mathrm{rig}}\right)_{\mathbb{C}_{p}}=\left(\mathfrak{F}_{\mathcal{O}_{\mathbb{C}_{p}}}\right)^{\mathrm{rig}}. \end{align*} $$
Let
$\mathcal {U}=\mathrm {Sp}(A)$
be a reduced
$\mathbb {C}_{p}$
-affinoid variety. From [Reference Bosch, Lütkebohmert and Raynaud11, Theorem 1.2] and [Reference Abbes1, Proposition 1.10.2(iii)], we see that
$A^{\circ }$
is an admissible formal
$\mathcal {O}_{\mathbb {C}_{p}}$
-algebra. Put
$\mathfrak {U}=\mathrm {Spf}(A^{\circ })$
. For any morphism
$\mathcal {U}\to \mathcal {W}_{\mathbb {C}_{p}}$
or
$\mathcal {U}\to \mathcal {W}^{G}_{\mathbb {C}_{p}}$
, we have an associated character
$\kappa ^{\mathcal {U}}$
or
$\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )$
and a notion of n-analyticity defined in the same way as before. Consider the base extensions
$$ \begin{align*} \pi_{w,\mathcal{O}_{\mathbb{C}_{p}}}: \mathfrak{IW}_{w,\mathcal{O}_{\mathbb{C}_{p}}}^{+} \overset{\gamma_{w,\mathcal{O}_{\mathbb{C}_{p}}}}{\to} \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}),\mu_{N}, \mathfrak{c})\left(\underline{v}\right)_{\mathcal{O}_{\mathbb{C}_{p}}}\overset{h_{n,\mathcal{O}_{\mathbb{C}_{p}}}}{\to} \bar{\mathfrak{M}}(\mu_{N},\mathfrak{c})\left(\underline{v}\right)_{\mathcal{O}_{\mathbb{C}_{p}}} \end{align*} $$
of the maps
$\gamma _{w}$
,
$h_{n}$
, and
$\pi _{w}$
. Then for any n-analytic morphism
$\mathcal {U}\to \mathcal {W}_{\mathbb {C}_{p}}$
, we can define the sheaves
$$ \begin{align*} \Omega^{\kappa^{\mathcal{U}}}=\left(\pi^{\mathrm{rig}}_{w,\mathbb{C}_{p}}\right)_{*}\left(\mathcal{O}_{\mathcal{IW}_{w,\mathbb{C}_{p}}^{+}\times \mathcal{U}}\right)\left[-\kappa^{\mathcal{U}}\right],\qquad \mathbf{\Omega}^{\kappa^{\mathcal{U}}}=\left(\pi_{w,\mathcal{O}_{\mathbb{C}_{p}}}\right)_{*}\left(\mathcal{O}_{\mathfrak{IW}_{w,\mathcal{O}_{\mathbb{C}_{p}}}^{+}\times \mathfrak{U}}\right)\left[-\boldsymbol{\kappa}^{\mathcal{U}}\right] \end{align*} $$
such that
$\Omega ^{\kappa ^{\mathcal {U}}}=\left (\mathbf {\Omega }^{\kappa ^{\mathcal {U}}}\right )^{\mathrm {rig}}$
is an invertible
$\mathcal {O}_{\bar {\mathcal {M}}\left (\mu _{N},\mathfrak {c}\right )(v)_{\mathbb {C}_{p}}\times \mathcal {U}}$
-module, as before. [Reference Abbes1, Propositions 1.9.14 and 1.10.2(iii)] implies that
$\mathbf {\Omega }^{\kappa ^{\mathcal {U}}}$
is coherent and that its restriction to
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{\mathcal {O}_{\mathbb {C}_{p}}}$
is invertible: the latter follows from a similar argument to the proof of [Reference Mumford35, §7, Proposition 2] combined with the fact that
$h_{n,\mathcal {O}_{\mathbb {C}_{p}}}$
is a
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
-torsor over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{\mathcal {O}_{\mathbb {C}_{p}}}$
. Using
$\Omega ^{\kappa ^{\mathcal {U}}}$
, we define
$M\left (\mu _{N},\mathfrak {c},\kappa ^{\mathcal {U}}\right )\left (\underline {v}\right )$
and its variants in the same way as in the case over K.
For any reduced K-affinoid variety
$\mathcal {V}$
and any n-analytic morphism
$\mathcal {V}\to \mathcal {W}$
, consider the base extension
$\mathcal {V}_{\mathbb {C}_{p}}\to \mathcal {W}_{\mathbb {C}_{p}}$
and the associated character
$\kappa ^{\mathcal {V}_{\mathbb {C}_{p}}}$
. Then we can show that there exist natural isomorphisms
$$ \begin{align} \left(\Omega^{\kappa^{\mathcal{V}}}\right)_{\mathbb{C}_{p}}\simeq \Omega^{\kappa^{\mathcal{V}_{\mathbb{C}_{p}}}},\qquad \Omega^{\kappa^{\mathcal{V}}}(-D)_{\mathbb{C}_{p}}\simeq \Omega^{\kappa^{\mathcal{V}_{\mathbb{C}_{p}}}}(-D) \end{align} $$
in the same way as the proof of [Reference Andreatta, Iovita and Pilloni4, Proposition 3.14]. Similarly, for any morphism
$f:\mathcal {U}^{\prime }\to \mathcal {U}$
of reduced
$\mathbb {C}_{p}$
-affinoid varieties, we have natural isomorphisms
$$ \begin{align} f^{*}\Omega^{\kappa^{\mathcal{U}}}\simeq \Omega^{\kappa^{\mathcal{U}^{\prime}}},\qquad f^{*}\left(\Omega^{\kappa^{\mathcal{U}}}(-D)\right)\simeq \Omega^{\kappa^{\mathcal{U}^{\prime}}}(-D). \end{align} $$
Let
$\bar {M}^{*}(\mu _{N},\mathfrak {c})$
be the minimal compactification of
$M(\mu _{N},\mathfrak {c})$
. We have a natural proper map
$$ \begin{align*} \bar{M}(\mu_{N},\mathfrak{c})\to \bar{M}^{*}(\mu_{N},\mathfrak{c}). \end{align*} $$
Note that a sufficiently large power of the usual Hasse invariant can be considered as a global section of an ample invertible sheaf on
$\bar {M}^{*}(\mu _{N},\mathfrak {c})$
. Let
$\bar {\mathfrak {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
be the normal admissible formal scheme defined similarly to
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
using
$\bar {M}^{*}(\mu _{N},\mathfrak {c})$
instead of
$\bar {M}(\mu _{N},\mathfrak {c})$
. Let
$\bar {\mathcal {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
be its Raynaud generic fiber. By the foregoing ampleness property, we see that
$\bar {\mathcal {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
is a K-affinoid variety. We also have proper morphisms
$$ \begin{align*} \rho &: \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v_{\mathrm{tot}}) \to \bar{\mathfrak{M}}^{*}(\mu_{N},\mathfrak{c})(v_{\mathrm{tot}}),\\ \rho^{\mathrm{rig}} &: \bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})\to \bar{\mathcal{M}}^{*}(\mu_{N},\mathfrak{c})(v_{\mathrm{tot}}). \end{align*} $$
By the base extension, these induce proper morphisms
$$ \begin{align*} \rho_{\mathcal{O}_{\mathbb{C}_{p}}} &: \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})_{\mathcal{O}_{\mathbb{C}_{p}}} \to \bar{\mathfrak{M}}^{*}(\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})_{\mathcal{O}_{\mathbb{C}_{p}}},\\ \rho^{\mathrm{rig}}_{\mathbb{C}_{p}} &: \bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})_{\mathbb{C}_{p}}\to \bar{\mathcal{M}}^{*}(\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})_{\mathbb{C}_{p}}. \end{align*} $$
Lemma 3.11. Let
$\mathcal {V}$
be a reduced K-affinoid variety and let
$\mathcal {V}\to \mathcal {W}^{G}$
be an n-analytic morphism. Then the natural base change map
$$ \begin{align*} (\rho\times 1)_{*} \left(\mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)\right)_{\mathcal{O}_{\mathbb{C}_{p}}}\to \left(\rho_{\mathcal{O}_{\mathbb{C}_{p}}}\times 1\right)_{*}\left(\mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)_{\mathcal{O}_{\mathbb{C}_{p}}}\right) \end{align*} $$
is an isomorphism. Moreover, we have
$$ \begin{align*} R^{q} \left(\rho_{\mathcal{O}_{\mathbb{C}_{p}}}\times 1\right)_{*}\left(\mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)_{\mathcal{O}_{\mathbb{C}_{p}}}\right)=0 \end{align*} $$
for any
$q>0$
.
Proof. It is enough to show the claim formal-locally. Put
$\mathcal {V}=\mathrm {Sp}(A)$
and
$\mathfrak {V}=\mathrm {Spf}(A^{\circ })$
. Let
$\mathfrak {Y}$
be a formal affine open subscheme of
$\bar {\mathfrak {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
and put
$\mathfrak {X}=\rho ^{-1}(\mathfrak {Y})$
. Since
$\rho $
is proper of finite presentation and
$\mathbf {\Omega }^{\kappa ^{\mathcal {V}}}(-D)$
is coherent, [Reference Abbes1, (2.11.8.1)] implies that the restriction
$$ \begin{align*} R^{q}(\rho\times 1)_{*} \left(\mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)\right)\rvert_{\mathfrak{Y}\times\mathfrak{V}} \end{align*} $$
is the coherent sheaf associated to the
$\mathcal {O}(\mathfrak {Y}\times \mathfrak {V})$
-module
$$ \begin{align*} H^{q}\left(\mathfrak{X}\times \mathfrak{V},\mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)\right). \end{align*} $$
By [Reference Andreatta, Iovita and Pilloni4, Corollary 3.20], we have
$H^{q}\left (\mathfrak {X}\times \mathfrak {V},\mathbf {\Omega }^{\kappa ^{\mathcal {V}}}(-D)\right )=0$
for any
$q>0$
.
Since
$\mathfrak {X}$
is quasi-compact, we can take a finite covering
$\mathfrak {X}=\bigcup _{i=1}^{r} \mathfrak {X}_{i}$
by formal affine open subschemes
$\mathfrak {X}_{i}$
. Consider the Čech complex for the coherent sheaf
$\mathbf {\Omega }^{\kappa ^{\mathcal {V}}}(-D)$
$$ \begin{align*} 0 \to H^{0}\left(\mathfrak{X}\times \mathfrak{V}, \mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)\right)\to C^{0}\left(\mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)\right)\to C^{1}\left(\mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)\right)\to\dotsb \end{align*} $$
with respect to the covering
$\mathfrak {X}\times \mathfrak {V} =\bigcup _{i=1}^{r} \mathfrak {X}_{i}\times \mathfrak {V}$
, which is exact by the foregoing vanishing. From the definition, we see that the sheaf
$\mathbf {\Omega }^{\kappa ^{\mathcal {V}}}(-D)$
is flat over
$\mathcal {O}_{K}$
and each
$\mathcal {O}_{K}$
-module
$C^{q}\left (\mathbf {\Omega }^{\kappa ^{\mathcal {V}}}(-D)\right )$
is also flat. By taking modulo
$p^{n}$
, tensoring
$\mathcal {O}_{\mathbb {C}_{p}}$
, and taking the inverse limit, we see that the sequence is exact even after taking
$-\hat {\otimes }_{\mathcal {O}_{K}}\mathcal {O}_{\mathbb {C}_{p}}$
. This means that the Čech complex for the coherent sheaf
$\mathbf {\Omega }^{\kappa ^{\mathcal {V}}}(-D)_{\mathcal {O}_{\mathbb {C}_{p}}}$
with respect to the formal open covering
$\mathfrak {X}_{\mathcal {O}_{\mathbb {C}_{p}}}\times \mathfrak {V}_{\mathcal {O}_{\mathbb {C}_{p}}}=\bigcup _{i=1}^{r} \mathfrak {X}_{i,\mathcal {O}_{\mathbb {C}_{p}}}\times \mathfrak {V}_{\mathcal {O}_{\mathbb {C}_{p}}}$
is exact except the zeroth degree. Taking the zeroth cohomology gives an isomorphism
$$ \begin{align*} H^{0}\left(\mathfrak{X}\times \mathfrak{V}, \mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)\right)\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{\mathbb{C}_{p}}\to H^{0}\left(\mathfrak{X}_{\mathcal{O}_{\mathbb{C}_{p}}}\times \mathfrak{V}_{\mathcal{O}_{\mathbb{C}_{p}}}, \mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)_{\mathcal{O}_{\mathbb{C}_{p}}}\right) \end{align*} $$
and the qth cohomology for
$q>0$
gives
$$ \begin{align*} H^{q}\left(\mathfrak{X}_{\mathcal{O}_{\mathbb{C}_{p}}}\times \mathfrak{V}_{\mathcal{O}_{\mathbb{C}_{p}}}, \mathbf{\Omega}^{\kappa^{\mathcal{V}}}(-D)_{\mathcal{O}_{\mathbb{C}_{p}}}\right)=0. \end{align*} $$
This concludes the proof.
Lemma 3.12. Let
$\mathcal {V}$
be a reduced K-affinoid variety and let
$\mathcal {V}\to \mathcal {W}^{G}$
be an n-analytic morphism. Then the natural map
$$ \begin{align*} S^{G}\left(\mu_{N},\left(\nu^{\mathcal{V}},w^{\mathcal{V}}\right)\right)(v_{\mathrm{tot}})\hat{\otimes}_{K} \mathbb{C}_{p} \to S^{G}\left(\mu_{N},\left(\nu^{\mathcal{V}_{\mathbb{C}_{p}}},w^{\mathcal{V}_{\mathbb{C}_{p}}}\right)\right)(v_{\mathrm{tot}}) \end{align*} $$
is an isomorphism.
Proof. Put
$\mathcal {V}=\mathrm {Sp}(A)$
. By taking the Raynaud generic fibers and [Reference Abbes1, Propositions 4.7.23 and 4.7.36], we see from Lemma 3.11 that the base-change map
$$ \begin{align*} \left(\rho^{\mathrm{rig}} \times 1\right)_{*} \left(\Omega^{\kappa^{\mathcal{V}}}(-D)\right)_{\mathbb{C}_{p}}\to \left(\rho_{\mathbb{C}_{p}}\times 1\right)_{*}\left(\Omega^{\kappa^{\mathcal{V}}}(-D)_{\mathbb{C}_{p}}\right) \end{align*} $$
is an isomorphism. By formula (3.9), the latter sheaf is isomorphic to the sheaf
$\left (\rho _{\mathbb {C}_{p}}\times 1\right )_{*}\left (\Omega ^{\kappa ^{\mathcal {V}_{\mathbb {C}_{p}}}}(-D)\right )$
. Since
$\bar {\mathcal {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})_{\mathbb {C}_{p}}\times \mathcal {V}_{\mathbb {C}_{p}}$
is a
$\mathbb {C}_{p}$
-affinoid variety, taking global sections yields an isomorphism
$$ \begin{align} &H^{0}\left(\bar{\mathcal{M}}(\Gamma_{1}(p^{n}), \mu_{N},\mathfrak{c})(v_{\mathrm{tot}})\times \mathcal{V},\Omega^{\kappa^{\mathcal{V}}}(-D)\right)\hat{\otimes}_{K}\mathbb{C}_{p}\to \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad H^{0}\left(\bar{\mathcal{M}}(\Gamma_{1}(p^{n}), \mu_{N},\mathfrak{c})(v_{\mathrm{tot}})_{\mathbb{C}_{p}}\times \mathcal{V}_{\mathbb{C}_{p}},\Omega^{\kappa^{\mathcal{V}_{\mathbb{C}_{p}}}}(-D)\right). \end{align} $$
Taking the
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
-equivariant part and the
$\Delta $
-fixed part, we obtain the lemma.
Lemma 3.13. Let
$\mathcal {V}=\mathrm {Sp}(A)$
be a reduced K-affinoid variety and let
$\mathcal {V}\to \mathcal {W}^{G}$
be an n-analytic morphism. Let x be an element of
$\mathcal {V}(\mathbb {C}_{p})$
and let
$x^{*}:A \to \mathbb {C}_{p}$
be the ring homomorphism defined by x. Suppose that the maximal ideal
$m_{x}$
of
$A_{\mathbb {C}_{p}}=A\hat {\otimes }_{K} \mathbb {C}_{p}$
corresponding to x is generated by a regular sequence. Put
$(\nu ,w)=\left (\nu ^{\mathcal {V}}(x),w^{\mathcal {V}}(x)\right )$
. Then the specialization map
$$ \begin{align*} S^{G}\left(\mu_{N}, \left(\nu^{\mathcal{V}},w^{\mathcal{V}}\right)\right)(v_{\mathrm{tot}})\hat{\otimes}_{A,x^{*}}\mathbb{C}_{p}\to S^{G}(\mu_{N}, (\nu,w))(v_{\mathrm{tot}}) \end{align*} $$
is an isomorphism.
Proof. This is essentially proved in [Reference Andreatta, Iovita and Pilloni4, Proposition 3.23]. Put
$\kappa ^{\mathcal {V}}=k\left (\nu ^{\mathcal {V}},w^{\mathcal {V}}\right )$
and
$\kappa =k(\nu ,w)$
. By the assumption on
$m_{x}$
, we have the Koszul resolution
$$ \begin{align*} 0\to A_{\mathbb{C}_{p}} \to A^{n_{r}}_{\mathbb{C}_{p}} \to \dotsb \to A^{n_{1}}_{\mathbb{C}_{p}}\to A_{\mathbb{C}_{p}} \to A_{\mathbb{C}_{p}}/m_{x} \to 0 \end{align*} $$
with some nonnegative integers
$n_{1},\dotsc ,n_{r}$
, which induces a finite resolution of the sheaf
$(1\times x)_{*}(\Omega ^{\kappa }(-D))$
by finite direct sums of
$\Omega ^{\kappa ^{\mathcal {V}}}(-D)_{\mathbb {C}_{p}}$
. By Lemma 3.11, the push-forward of this resolution by the map
$\rho ^{\mathrm {rig}}_{\mathbb {C}_{p}}\times 1$
is exact. Since
$\bar {\mathcal {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})_{\mathbb {C}_{p}}\times \mathcal {V}_{\mathbb {C}_{p}}$
is a
$\mathbb {C}_{p}$
-affinoid variety, the sequence obtained by taking global sections is also exact. This and formula (3.11) yield isomorphisms
$$ \begin{align*} H^{0}\left(\bar{\mathcal{M}}\vphantom{\Omega^{\kappa^{\mathcal{V}}}}\right.&\left.(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c}\vphantom{\Omega^{\kappa^{\mathcal{V}}}})(v_{\mathrm{tot}})\times \mathcal{V}, \Omega^{\kappa^{\mathcal{V}}}(-D)\right)\hat{\otimes}_{A,x^{*}}\mathbb{C}_{p}\\ &\simeq H^{0}\left(\bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})_{\mathbb{C}_{p}}\times \mathcal{V}_{\mathbb{C}_{p}}, \Omega^{\kappa^{\mathcal{V}_{\mathbb{C}_{p}}}}(-D)\right)\hat{\otimes}_{A_{\mathbb{C}_{p}},x^{*}}\mathbb{C}_{p}\\ &\simeq H^{0}\left(\bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})_{\mathbb{C}_{p}}, \Omega^{\kappa}(-D)\right). \end{align*} $$
Taking the
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
-equivariant part and the
$\Delta $
-fixed part shows the lemma.
We can naturally extend the Hecke operators over
$\mathbb {C}_{p}$
: Let
$\mathcal {U}$
be a smooth
$\mathbb {C}_{p}$
-affinoid variety and let
$\mathcal {U}\to \mathcal {W}^{G}_{\mathbb {C}_{p}}$
be an n-analytic morphism. Consider the base extension of the isomorphism
$\pi _{\mathfrak {l}}$
$$ \begin{align*} \pi_{\mathfrak{l}, \mathbb{C}_{p}}: p_{2}^{*} \mathcal{IW}_{w,\mathfrak{l}\mathfrak{c}}^{+}(v)_{\mathbb{C}_{p}} \simeq p_{1}^{*}\mathcal{IW}_{w,\mathfrak{c}}^{+}(v)_{\mathbb{C}_{p}}, \end{align*} $$
which defines an isomorphism
$$ \begin{align*} \pi_{\mathfrak{l},\mathbb{C}_{p}}^{*}: p_{1}^{*}\left(\Omega^{\kappa^{\mathcal{U}}}\right)\simeq p_{2}^{*}\left(\Omega^{\kappa^{\mathcal{U}}}\right). \end{align*} $$
We define the Hecke operator
$T_{\mathfrak {l}}$
over
$\mathbb {C}_{p}$
for
$(\mathfrak {l},p)=1$
by
$$ \begin{align*} H^{0}\left(\mathcal{M}(\mu_{N},\mathfrak{l}\mathfrak{c})\left(\underline{v}\right)_{\mathbb{C}_{p}}\times \mathcal{U},\Omega^{\kappa^{\mathcal{U}}}\right)& \overset{p_{2}^{*}}{\to} H^{0}\left(\mathcal{Y}^{\prime}_{\mathfrak{c},\mathfrak{l}}\left(\underline{v}\right)_{\mathbb{C}_{p}}\times \mathcal{U},p_{2}^{*}\Omega^{\kappa^{\mathcal{U}}}\right)\\ & \overset{\left(\pi_{\mathfrak{l},\mathbb{C}_{p}}^{*}\right)^{-1}}{\to} H^{0}\left(\mathcal{Y}^{\prime}_{\mathfrak{c},\mathfrak{l}}\left(\underline{v}\right)_{\mathbb{C}_{p}}\times \mathcal{U},p_{1}^{*}\Omega^{\kappa^{\mathcal{U}}}\right)\\ & \overset{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{l})^{-1}\mathrm{Tr}_{p_{1}}}{\to} H^{0}\left(\mathcal{M}(\mu_{N},\mathfrak{c})(\underline{v})_{\mathbb{C}_{p}}\times \mathcal{U},\Omega^{\kappa^{\mathcal{U}}}\right). \end{align*} $$
Similarly, we have Hecke operators
$T_{\mathfrak {l}}$
for
$(\mathfrak {l},p)\neq 1$
and
$S_{\mathfrak {l}}$
over
$\mathbb {C}_{p}$
. We can show that they are compatible with the Hecke operators over K and that the specialization map in Lemma 3.13 is
$\mathbf {T}$
-linear.
4 q-Expansion principle
In this section, we study the q-expansion map for arithmetic overconvergent Hilbert modular forms. We fix once and for all a representative
$$ \begin{align*} \left[\mathrm{Cl}^{+}(F)\right]^{(p)}=\left\{\mathfrak{c}_{1}=\mathfrak{o},\mathfrak{c}_{2},\dotsc, \mathfrak{c}_{h^{+}}\right\} \end{align*} $$
of the strict class group
$\mathrm {Cl}^{+}(F)$
such that every
$\mathfrak {c}_{i}$
is prime to p. For any smooth
$\mathbb {C}_{p}$
-affinoid variety
$\mathcal {U}$
, any n-analytic map
$\mathcal {U}\to \mathcal {W}^{G}_{\mathbb {C}_{p}}$
, and any
$v\in \mathbb {Q}\cap [0,B_{n})$
, we have isomorphisms
$$ \begin{align} \begin{aligned} M^{G}\left(\mu_{N}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v)&\simeq \bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} M^{G}\left(\mu_{N},\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v),\\ S^{G}\left(\mu_{N}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v)&\simeq \bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} S^{G}\left(\mu_{N},\mathfrak{c}, \left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v) \end{aligned} \end{align} $$
by which we identify both sides. For any element
$f\in M^{G}\left (\mu _{N}, \left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v)$
, we write
$(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
for the image of f by these isomorphisms. We say that f is an eigenform if it is an eigenvector for any element of
$\mathbf {T}$
.
4.1 q-Expansion of overconvergent modular forms
For any nonzero fractional ideal
$\mathfrak {c}$
of F, let us consider an unramified cusp
$(\mathfrak {a},\mathfrak {b},\phi )$
of
$M(\mu _{N},\mathfrak {c})$
as in §3.1.2. Using any polyhedral cone decomposition
$\mathscr {C}\in \mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
of
$F^{*,+}_{\mathbb {R}}$
, we have the
$\hat {I}_{\sigma }$
-adically complete ring
$\hat {R}_{\sigma }$
and the semi-abelian scheme
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
over
$\bar {S}_{\sigma }=\mathrm {Spec}\left (\hat {R}_{\sigma }\right )$
for any
$\sigma \in \mathscr {C}$
.
Let
$\breve {S}_{\sigma }=\mathrm {Spf}\left (\breve {R}_{\sigma }\right )$
be the
$\left (p,\hat {I}_{\sigma }\right )$
-adic formal completion of
$\hat {S}_{\sigma }$
. The smoothness assumption on
$\mathscr {C}$
implies that there exists a basis
$\xi _{1},\dotsc ,\xi _{g}$
of the
$\mathbb {Z}$
-module
$\mathfrak {a}\mathfrak {b}$
satisfying
$$ \begin{align*} (\mathfrak{a}\mathfrak{b})\cap \sigma^{\vee}=\mathbb{Z}_{\geq 0}\xi_{1}+\dotsb+\mathbb{Z}_{\geq 0}\xi_{r}+\mathbb{Z}\xi_{r+1}+\dotsb+\mathbb{Z}\xi_{g} \end{align*} $$
with some r. For any ring B, we write
$$ \begin{align*} B\left[X_{\leq r}, X^{\pm}_{>r}\right]:=B\left[X_{1},\dotsc,X_{r},X_{r+1}^{\pm},\dotsc,X_{g}^{\pm}\right]. \end{align*} $$
For any extension
$L/K$
of complete valuation fields, we denote the p-adic completion of
$\mathcal {O}_{L}\left [X_{\leq r}, X^{\pm }_{>r}\right ]$
by
$\mathcal {O}_{L}\left \langle X_{\leq r}, X^{\pm }_{>r}\right \rangle $
and put
$$ \begin{align*} L\left\langle X_{\leq r}, X^{\pm}_{>r} \right\rangle=\mathcal{O}_{L}\left\langle X_{\leq r}, X^{\pm}_{>r}\right\rangle[1/p]. \end{align*} $$
Then the
$\mathcal {O}_{K}$
-algebra
$\hat {R}_{\sigma }$
is isomorphic to the completion of the ring
$\mathcal {O}_{K}\left [X_{\leq r}, X^{\pm }_{>r}\right ]$
with respect to the principal ideal
$(X_{1}\dotsm X_{r})$
via the map
$X_{i}\mapsto q^{\xi _{i}}$
, and the ring
$\breve {R}_{\sigma }$
is isomorphic to the p-adic completion of
$\hat {R}_{\sigma }$
. Hence the ring
$\breve {R}_{\sigma }$
is normal and the formal scheme
$\breve {S}_{\sigma }$
is an object of the category
$\mathrm {FS} {}_{\mathcal {O}_{K}}$
of [Reference de Jong19, Definition 7.0.1]. In fact, the ring
$\breve {R}_{\sigma }$
is isomorphic to the ring
$$ \begin{align} \mathcal{O}_{K}\left\langle X_{\leq r}, X^{\pm}_{>r}\right\rangle [[Z]]/(Z-X_{1}\dotsm X_{r}). \end{align} $$
Moreover, since the natural map
$$ \begin{align*} &\mathcal{O}_{K,m}\left[X_{\leq r}, X^{\pm}_{>r}\right]/(X_{1}\dotsm X_{r})^{n} \to \\ &\qquad\qquad\qquad\qquad\qquad\qquad\mathcal{O}_{K,m}\left[X_{r+1}^{\pm},\dotsc,X_{g}^{\pm}\right][[X_{1},\dotsc,X_{r}]]/(X_{1}\dotsm X_{r})^{n} \end{align*} $$
is injective for any positive integer m, by taking the limit we may identify the rings
$\hat {R}_{\sigma }$
and
$\breve {R}_{\sigma }$
with
$\mathcal {O}_{K}$
-subalgebras of the
$\mathcal {O}_{K}$
-algebra
$$ \begin{align*} \mathcal{O}_{K}\left\langle q^{\pm\xi_{r+1}},\dotsc, q^{\pm\xi_{g}}\right\rangle\left[\left[q^{\xi_{1}},\dotsc,q^{\xi_{r}}\right]\right]. \end{align*} $$
We denote by
$\breve {S}_{\sigma }^{\mathrm {rig}}$
the Berthelot generic fiber of
$\breve {S}_{\sigma }$
. Similarly, we denote by
$\breve {S}_{\mathscr {C}}$
and
$\breve {S}^{\mathrm {rig}}_{\mathscr {C}}$
the formal completion of
$\hat {S}_{\mathscr {C}}$
along the boundary of the special fiber and its Berthelot generic fiber. From the definition, we have formal open and admissible coverings
$$ \begin{align*} \breve{S}_{\mathscr{C}}=\bigcup_{\sigma\in \mathscr{C}} \breve{S}_{\sigma},\qquad \breve{S}^{\mathrm{rig}}_{\mathscr{C}}=\bigcup_{\sigma\in \mathscr{C}} \breve{S}^{\mathrm{rig}}_{\sigma}. \end{align*} $$
Since the quotient of
$\hat {S}_{\mathscr {C}}$
by the action of
$U_{N}$
is obtained by a regluing, so is the quotient
$\breve {S}_{\mathscr {C}}/U_{N}$
, and this coincides with the formal completion of
$\hat {S}_{\mathscr {C}}/U_{N}$
along the boundary of the special fiber.
Consider the case
$\mathscr {C}=\mathscr {C}(\mathfrak {a},\mathfrak {b})$
. Since the map
$\bar {S}_{\sigma }\to \bar {M}(\mu _{N},\mathfrak {c})$
defined using
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
induces an open immersion
$$ \begin{align*} \coprod \hat{S}_{\mathscr{C}(\mathfrak{a},\mathfrak{b})}/U_{N} \to \bar{M}(\mu_{N},\mathfrak{c})\rvert_{D}^{\wedge} \end{align*} $$
to the formal completion of
$\bar {M}(\mu _{N},\mathfrak {c})\rvert _{D}^{\wedge }$
of
$\bar {M}(\mu _{N},\mathfrak {c})$
along the boundary divisor D, taking the formal completion we obtain an open immersion
$$ \begin{align*} \coprod \breve{S}_{\mathscr{C}(\mathfrak{a},\mathfrak{b})}/U_{N} \to \bar{\mathfrak{M}}(\mu_{N},\mathfrak{c})\rvert_{D_{k}}^{\wedge} \end{align*} $$
to the formal completion
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\rvert _{D_{k}}^{\wedge }$
of
$\bar {M}(\mu _{N},\mathfrak {c})$
along the boundary
$D_{k}$
of the special fiber. Let
$\mathrm {sp}: \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\to \bar {M}(\mu _{N},\mathfrak {c})_{k}$
be the specialization map with respect to
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
. Then [Reference de Jong19, Lemma 7.2.5] implies
$\left (\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\rvert _{D_{k}}^{\wedge }\right )^{\mathrm {rig}}=\mathrm {sp}^{-1}(D_{k})$
.
Let
$\breve {S}^{\mathrm {rig}}_{\sigma , \mathbb {C}_{p}}$
and
$\breve {S}^{\mathrm {rig}}_{\mathscr {C}, \mathbb {C}_{p}}$
be the base extensions to
$\mathrm {Sp}\left (\mathbb {C}_{p}\right )$
of
$\breve {S}^{\mathrm {rig}}_{\sigma }$
and
$\breve {S}^{\mathrm {rig}}_{\mathscr {C}}$
, respectively. Note that
$\breve {S}^{\mathrm {rig}}_{\sigma ,\mathbb {C}_{p}}$
can be identified with the rigid analytic variety over
$\mathbb {C}_{p}$
whose set of
$\mathbb {C}_{p}$
-points is
$$ \begin{align} \left\{\left(x_{1},\dotsc,x_{g}\right)\in \mathbb{C}_{p}^{g}\ \middle| \begin{array}{cc} x_{i}\in \mathcal{O}_{\mathbb{C}_{p}}\ (i\leq r),\ x_{i}\in \mathcal{O}^{\times}_{\mathbb{C}_{p}}\ (i> r), \\ x_{1}\dotsm x_{r}\in m_{\mathbb{C}_{p}} \end{array} \!\!\!\!\!\right\} \end{align} $$
for r as before. Then, with the notation of [Reference Conrad18, Theorem 3.1.5], we have
$$ \begin{align*} \left(\breve{S}_{\sigma}\right)^{\mathrm{rig}}_{/\mathbb{C}_{p}}=\breve{S}^{\mathrm{rig}}_{\sigma, \mathbb{C}_{p}},\qquad \left(\breve{S}_{\mathscr{C}}\right)^{\mathrm{rig}}_{/\mathbb{C}_{p}}=\breve{S}^{\mathrm{rig}}_{\mathscr{C}, \mathbb{C}_{p}}. \end{align*} $$
Since the functor
$(-)^{\mathrm {rig}}_{/\mathbb {C}_{p}}$
sends formal open immersions to open immersions and formal open coverings to admissible coverings, each
$\breve {S}^{\mathrm {rig}}_{\sigma ,\mathbb {C}_{p}}$
is an admissible open subset of
$\breve {S}^{\mathrm {rig}}_{\mathscr {C},\mathbb {C}_{p}}$
such that
$\breve {S}^{\mathrm {rig}}_{\mathscr {C},\mathbb {C}_{p}}=\bigcup _{\sigma \in \mathscr {C}} \breve {S}^{\mathrm {rig}}_{\sigma ,\mathbb {C}_{p}}$
is an admissible covering. Moreover, we have
$$ \begin{align*} \left(\breve{S}_{\mathscr{C}}/U_{N}\right)^{\mathrm{rig}}_{/\mathbb{C}_{p}}=\breve{S}^{\mathrm{rig}}_{\mathscr{C},\mathbb{C}_{p}}/U_{N}. \end{align*} $$
Note that the formation of the tube
$\mathrm {sp}^{-1}(D_{k})$
is compatible with the base extension to
$\mathbb {C}_{p}$
[Reference Berthelot9, Proposition 1.1.13]. Thus, for
$\mathscr {C}=\mathscr {C}(\mathfrak {a},\mathfrak {b})$
, we obtain maps
$$ \begin{align} \coprod_{\sigma\in \mathscr{C}} \breve{S}^{\mathrm{rig}}_{\sigma,\mathbb{C}_{p}} \to \breve{S}_{\mathscr{C},\mathbb{C}_{p}}^{\mathrm{rig}}/U_{N} \to \bar{\mathcal{M}}(\mu_{N},\mathfrak{c})_{\mathbb{C}_{p}}, \end{align} $$
where the first map is a surjective local isomorphism and the second is an open immersion factoring through
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(0)_{\mathbb {C}_{p}}$
.
We denote by
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\breve {S}_{\sigma , \mathcal {O}_{\mathbb {C}_{p}}}$
, and
$\breve {S}_{\mathscr {C}, \mathcal {O}_{\mathbb {C}_{p}}}$
the base extensions to
$\mathrm {Spf}\left (\mathcal {O}_{\mathbb {C}_{p}}\right )$
of
$\breve {R}_{\sigma }$
,
$\breve {S}_{\sigma }$
, and
$\breve {S}_{\mathscr {C}}$
, respectively. From the identification (4.2), we can show
$$ \begin{align} \breve{R}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}=\mathcal{O}_{\mathbb{C}_{p}}\left\langle X_{\leq r}, X^{\pm}_{>r}\right\rangle [[Z]]/(Z-X_{1}\dotsm X_{r}). \end{align} $$
Indeed, first note that the ring
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
is isomorphic to
$$ \begin{align} \varprojlim_{n\geq 0}\varprojlim_{m\geq 0} \mathcal{O}_{\mathbb{C}_{p},n}\left[X_{\leq r}, X^{\pm}_{>r},Z\right]/(Z-X_{1}\dotsm X_{r}, Z^{m}). \end{align} $$
Since the ring
$$ \begin{align*} \mathcal{O}_{\mathbb{C}_{p},n}\left[X_{\leq r}, X^{\pm}_{>r},Z\right]/(Z-X_{1}\dotsm X_{r}) \end{align*} $$
is Z-torsion free, its Z-adic completion is
$$ \begin{align*} \mathcal{O}_{\mathbb{C}_{p},n}\left[X_{\leq r}, X^{\pm}_{>r}\right][[Z]]/(Z-X_{1}\dotsm X_{r}). \end{align*} $$
Similarly, since an elementary argument shows that the ring
$$ \begin{align*} \mathcal{O}_{\mathbb{C}_{p}}\left[X_{\leq r}, X^{\pm}_{>r}\right][[Z]]/(Z-X_{1}\dotsm X_{r}) \end{align*} $$
is p-torsion free, taking the p-adic completion yields the claim. (The reason for this ad hoc proof is that in general we do not know if the completion is compatible with quotients for non-quasi-idyllic rings.)
Lemma 4.1. For any extension
$L/K$
of complete valuation fields with residue field
$k_{L}$
, the rings
$$ \begin{align*} \mathcal{O}_{L}\left\langle X_{\leq r}, X^{\pm}_{>r}\right\rangle [[Z]]/(Z-X_{1}\dotsm X_{r}),\qquad k_{L}\left[X_{\leq r}, X^{\pm}_{>r}\right] [[Z]]/(Z-X_{1}\dotsm X_{r}) \end{align*} $$
are integral domains. In particular, the ring
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
is an integral domain.
Proof. For the former ring, we can show that it is a subring of the ring
$$ \begin{align*} \breve{R}_{L}:=L\left\langle X_{\leq r}, X^{\pm}_{>r} \right\rangle [[Z]]/(Z-X_{1}\dotsm X_{r}). \end{align*} $$
It suffices to show that
$\breve {R}_{L}$
is an integral domain. Since the ring
$L\left \langle X_{\leq r}, X^{\pm }_{>r}\right \rangle $
is Noetherian and normal, the ring
$\breve {R}_{L}$
is also normal. Since
$\breve {R}_{L}$
is Z-adically complete and Z-torsion free and
$\mathrm {Spec}\left (\breve {R}_{L}/(Z)\right )$
is connected, we see that
$\mathrm {Spec}\left (\breve {R}_{L}\right )$
is also connected, and the lemma follows. We can show the assertion on the latter ring similarly.
From the description (4.3) of
$\breve {S}_{\sigma ,\mathbb {C}_{p}}^{\mathrm {rig}}$
, we see that there exists an inclusion
$$ \begin{align*} \mathcal{O}\left(\breve{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\right)=\breve{R}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\subseteq \mathcal{O}^{\circ}\left(\breve{S}^{\mathrm{rig}}_{\sigma,\mathbb{C}_{p}}\right). \end{align*} $$
By gluing, this yields an inclusion
$$ \begin{align} \mathcal{O}\left(\breve{S}_{\mathscr{C},\mathcal{O}_{\mathbb{C}_{p}}}\right) \subseteq \mathcal{O}^{\circ}\left(\breve{S}_{\mathscr{C},\mathbb{C}_{p}}^{\mathrm{rig}}\right). \end{align} $$
By the description (4.6) of the ring
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
, we have a natural inclusion
$$ \begin{align} \breve{R}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}} \subseteq \prod_{\xi\in \mathfrak{a}\mathfrak{b}} \mathcal{O}_{\mathbb{C}_{p}} q^{\xi}, \end{align} $$
which is compatible with the restriction map
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}} \to \breve {R}_{\sigma ^{\prime },\mathcal {O}_{\mathbb {C}_{p}}}$
for any
$\sigma $
and
$\sigma ^{\prime }$
such that
$\sigma ^{\prime }$
is a face of the closure
$\bar {\sigma }$
. Then we have an isomorphism
$$ \begin{align*} \mathcal{O}\left(\breve{S}_{\mathscr{C},\mathcal{O}_{\mathbb{C}_{p}}}\right)\simeq \bigcap_{\sigma\in \mathscr{C}} \breve{R}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}. \end{align*} $$
Note that if the dimension of the
$\mathbb {R}$
-vector space
$\mathrm {Span}_{\mathbb {R}}(\sigma )$
generated by the elements of
$\sigma $
is g, then we have
$$ \begin{align*} (\mathfrak{a}\mathfrak{b})\cap \sigma^{\vee}=\mathbb{Z}_{\geq 0} \xi_{1}\oplus\dotsb \oplus \mathbb{Z}_{\geq 0} \xi_{g} \end{align*} $$
with some
$\xi _{1},\dotsc ,\xi _{g}\in \mathfrak {a}\mathfrak {b}$
. Thus any element of
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
is a formal power series of
$q^{\xi _{1}},\dotsc ,q^{\xi _{g}}$
and the ring
$\mathcal {O}_{\mathbb {C}_{p}}\left [\left [q^{\xi }\mid \xi \in (\mathfrak {a}\mathfrak {b})\cap \sigma ^{\vee }\right ]\right ]$
can be identified with the subset
$$ \begin{align*} \left\{\left(a_{\xi} q^{\xi}\right)_{\xi \in \mathfrak{a}\mathfrak{b}}\in \prod_{\xi\in \mathfrak{a}\mathfrak{b}} \mathcal{O}_{\mathbb{C}_{p}} q^{\xi} \mid a_{\xi}=0\ \text{ for any } \xi\notin (\mathfrak{a}\mathfrak{b})\cap \sigma^{\vee} \right\}. \end{align*} $$
From the equality
$$ \begin{align*} (\mathfrak{a}\mathfrak{b})^{+}\cup \{0\}=\bigcap\left\{(\mathfrak{a}\mathfrak{b})\cap \sigma^{\vee}\mid \sigma\in\mathscr{C},\ \dim_{\mathbb{R}}(\mathrm{Span}_{\mathbb{R}}(\sigma))=g\right\}, \end{align*} $$
we have an inclusion
$$ \begin{align*} \mathcal{O}_{\mathbb{C}_{p}}\left[\left[q^{\xi} \mid \xi\in (\mathfrak{a}\mathfrak{b})^{+}\cup\{0\}\right]\right]\supseteq \mathcal{O}\left(\breve{S}_{\mathscr{C},\mathcal{O}_{\mathbb{C}_{p}}}\right). \end{align*} $$
On the other hand, if we identify
$$ \begin{align*} F_{\mathbb{R}}\simeq \prod_{\beta\in \mathrm{Hom}_{\mathbb{Q}\text{-alg.}}(F,\mathbb{R})} \mathbb{R},\quad x\otimes 1\mapsto (\beta(x))_{\beta}, \end{align*} $$
then every boundary
$\tau $
of
$\sigma ^{\vee }$
is outside the closure of the positive cone
$F_{\mathbb {R}}^{\times ,+}$
of
$F_{\mathbb {R}}$
. Hence for any positive real number
$\rho $
, the number of elements
$\xi $
of
$(\mathfrak {a}\mathfrak {b})^{+}$
such that the distance from
$\xi $
to
$\tau $
is less than
$\rho $
is finite. This implies that any element of
$\mathcal {O}_{\mathbb {C}_{p}}\left [\left [q^{\xi }\mid \xi \in (\mathfrak {a}\mathfrak {b})^{+}\cup \{0\}\right ]\right ]$
is contained in the completion of the ring
$$ \begin{align*} \mathcal{O}_{\mathbb{C}_{p}}\left[q^{\xi_{1}},\dotsc, q^{\xi_{r}},q^{\pm \xi_{r+1}},\dotsc, q^{\pm \xi_{g}}\right] \end{align*} $$
with respect to the
$q^{\xi _{1}}\dotsm q^{\xi _{r}}$
-adic topology. We can see that this completion is contained in
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
. Therefore, we obtain an identification
$$ \begin{align*} \mathcal{O}_{\mathbb{C}_{p}}\left[\left[q^{\xi}\mid \xi\in (\mathfrak{a}\mathfrak{b})^{+}\cup\{0\}\right]\right]\simeq \mathcal{O}\left(\breve{S}_{\mathscr{C},\mathcal{O}_{\mathbb{C}_{p}}}\right), \end{align*} $$
which is compatible with the inclusion (4.8).
Let
$\mathfrak {c}$
be any nonzero fractional ideal of F and let
$\mathfrak {a},\mathfrak {b}$
be fractional ideals satisfying
$\mathfrak {a}\mathfrak {b}^{-1}=\mathfrak {c}$
. Suppose
$\mathfrak {a}\subseteq \mathfrak {o}$
and
$(\mathfrak {a},Np)=1$
. Then the natural inclusion
$\mathfrak {o}\subseteq \mathfrak {a}^{-1}$
induces isomorphisms
$$ \begin{align*} \phi_{\mathfrak{a},\mathfrak{b}}: \mathfrak{a}^{-1}/N\mathfrak{a}^{-1} \simeq \mathfrak{o}/N\mathfrak{o},\qquad \phi^{\prime}_{\mathfrak{a},\mathfrak{b}}: \mathfrak{a}^{-1}/p^{n} \mathfrak{a}^{-1} \simeq \mathfrak{o}/p^{n}\mathfrak{o}. \end{align*} $$
Consider the unramified cusp
$\left (\mathfrak {a},\mathfrak {b},\phi _{\mathfrak {a},\mathfrak {b}}\right )$
of
$M(\mu _{N},\mathfrak {c})$
. Take
$\mathscr {C}\in \mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
and
$\sigma \in \mathscr {C}$
as before. By the construction of the Tate object, the map
$\phi ^{\prime }_{\mathfrak {a},\mathfrak {b}}$
yields a natural immersion
$\mathcal {D}_{F}^{-1}\otimes \mu _{p^{n}}\to \mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
over
$\bar {S}_{\sigma }$
, which induces an isomorphism
$$ \begin{align*} \omega_{\mathrm{Tate}_{\mathfrak{a},\mathfrak{b}}(q)}\otimes_{\mathcal{O}_{K}}\mathcal{O}_{K,n}\simeq \omega_{\mathcal{D}_{F}^{-1}\otimes \mu_{p^{n}}}. \end{align*} $$
Note that the map
$\mathrm {Tr}_{F/\mathbb {Q}}\otimes 1: \mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}\to \mathbb {G}_{\mathrm {m}}$
gives an element
$\left (\mathrm {Tr}_{F/\mathbb {Q}}\otimes 1\right )^{*}\frac {dT}{T}$
of the
$\mathcal {O}_{F}\otimes \mathcal {O}\left (\bar {S}_{\sigma }\right )$
-module
$\omega _{\mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}}\simeq \omega _{\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)}$
. By the pullback, we obtain a Tate object over
$\mathrm {Spec}\left (\breve {R}_{\sigma }\right )$
with a canonical invariant differential form which are compatible with those over
$\mathrm {Spec}\left (\breve {R}_{\tau }\right )$
for any
$\tau \in \mathscr {C}$
satisfying
$\tau \subseteq \bar {\sigma }$
.
We denote the p-adic completion of
$\bar {S}_{\sigma }$
by
$\tilde {S}_{\sigma }$
. We have
$\tilde {S}_{\sigma }=\mathrm {Spf}\left (\breve {R}_{\sigma }\right )$
, where we consider the p-adic topology on
$\breve {R}_{\sigma }$
. Its base extension to
$\mathcal {O}_{\mathbb {C}_{p}}$
is denoted by
$\tilde {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}=\mathrm {Spf}\left (\tilde {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}\right )$
. Here the affine algebra
$ \tilde {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
is the p-adic completion of the ring
$\breve {R}_{\sigma }\otimes _{\mathcal {O}_{K}} \mathcal {O}_{\mathbb {C}_{p}}$
.
The identity map
$\breve {R}_{\sigma }\to \breve {R}_{\sigma }$
is continuous if we consider the p-adic topology on the source and the
$\left (p,\hat {I}_{\sigma }\right )$
-adic topology on the target. Then, for the case of
$\mathscr {C}=\mathscr {C}(\mathfrak {a},\mathfrak {b})$
, its composite with the p-adic completion of the map
$\bar {S}_{\sigma }\to \bar {M}(\mu _{N},\mathfrak {c})$
gives a morphism of formal schemes
$\breve {S}_{\sigma } \to \tilde {S}_{\sigma }\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
, and also a morphism
$\breve {S}_{\mathscr {C}} \to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
by gluing. Since
$\breve {R}_{\sigma }$
is Noetherian, the moduli interpretation of
$\mathfrak {IW}_{w,\mathfrak {c}}^{+}(0)$
as in §3.3.1 is also valid for
$\breve {R}_{\sigma }$
. We have a commutative diagram
where the top horizontal arrow is the natural inclusion and the other horizontal arrows are induced by the map
$\mathrm {Tr}_{F/\mathbb {Q}}\otimes 1$
. Thus the moduli interpretation and the base extension give a morphism of formal schemes over
$\mathrm {Spf}\left (\mathcal {O}_{\mathbb {C}_{p}}\right )$
,
$$ \begin{align*} \tau_{\mathfrak{a},\mathfrak{b}}: \breve{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\to \tilde{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}} \to \mathfrak{IW}_{w,\mathfrak{c}}^{+}(0)_{\mathcal{O}_{\mathbb{C}_{p}}}. \end{align*} $$
By gluing, this defines a morphism
$\breve {S}_{\mathscr {C},\mathcal {O}_{\mathbb {C}_{p}}}\to \mathfrak {IW}^{+}_{w,\mathfrak {c}}(0)_{\mathcal {O}_{\mathbb {C}_{p}}}$
, which we also denote by
$\tau _{\mathfrak {a},\mathfrak {b}}$
.
Lemma 4.2. The natural map
$\tilde {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}} \to \breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
is injective. In particular, the ring
$\tilde {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
is an integral domain.
Proof. We have an isomorphism
$$ \begin{align} \tilde{R}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\simeq \varprojlim_{n} \varinjlim_{L/K} \mathcal{O}_{L,n}\left[X_{\leq r}, X_{>r}^{\pm}\right][[Z]]/(Z-X_{1}\dotsm X_{r}), \end{align} $$
where the direct limit is taken with respect to the directed set of finite extensions
$L/K$
in
$\bar {\mathbb {Q}}_{p}$
. Since the map
$$ \begin{align*} \mathcal{O}_{L,n}\left[X_{\leq r}, X_{>r}^{\pm}\right]/(X_{1}\dotsm X_{r})^{m} \to \mathcal{O}_{\mathbb{C}_{p},n}\left[X_{\leq r}, X_{>r}^{\pm}\right]/(X_{1}\dotsm X_{r})^{m} \end{align*} $$
is injective for any such
$L/K$
, the injectivity of the lemma follows from expression (4.6). Lemma 4.1 yields the last assertion.
For any finite extension
$L/K$
, we write the p-adic completion
$$ \begin{align*} \breve{R}_{\sigma} \hat{\otimes}_{\mathcal{O}_{K}} \mathcal{O}_{L}=\mathcal{O}_{L}\left\langle X_{\leq r}, X_{>r}^{\pm}\right\rangle [[Z]]/(Z-X_{1}\dotsm X_{r}) \end{align*} $$
also as
$\tilde {R}_{\mathcal {O}_{L}}$
. Let
$\pi _{L}$
be a uniformizer of L. By Lemma 4.1, the ring
$\tilde {R}_{\mathcal {O}_{L}}/(\pi _{L})$
is an integral domain. Since
$\tilde {R}_{\mathcal {O}_{L}}$
is normal, the localization
$\left (\tilde {R}_{\mathcal {O}_{L}}\right )_{\left (\pi _{L}\right )}$
is a discrete valuation ring with uniformizer
$\pi _{L}$
such that Z is invertible. Put
$\tilde {R}_{\infty }=\varinjlim _{L/K} \tilde {R}_{\mathcal {O}_{L}}$
and
$m_{\infty }=\varinjlim _{L/K} (\pi _{L})$
, where the direct limits are taken as before. Then the localization
$\left (\tilde {R}_{\infty }\right )_{m_{\infty }}=\varinjlim _{L/K} \left (\tilde {R}_{\mathcal {O}_{L}}\right )_{\left (\pi _{L}\right )}$
is a valuation ring. Let
$\mathcal {O}_{\mathcal {K}_{\sigma }}$
be its p-adic completion. By formula (4.9), the ring
$\tilde {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
coincides with the p-adic completion of
$\tilde {R}_{\infty }$
. Since the p-adic topology on
$\tilde {R}_{\infty }$
is induced by that on
$\left (\tilde {R}_{\infty }\right )_{m_{\infty }}$
, we obtain an injection
$\tilde {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}\to \mathcal {O}_{\mathcal {K}_{\sigma }}$
. This defines a morphism of p-adic formal schemes
$\mathrm {Spf}\left (\mathcal {O}_{\mathcal {K}_{\sigma }}\right )\to \tilde {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
for any
$\sigma \in \mathscr {C}(\mathfrak {a},\mathfrak {b})$
. In particular, we have the pullback of
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
over
$\mathrm {Spec}\left (\mathcal {O}_{\mathcal {K}_{\sigma }}\right )$
which is an HBAV. Since
$\mathcal {O}_{\mathcal {K}_{\sigma }}$
is quasi-idyllic, we have the moduli interpretation of any morphism
$\mathrm {Spf}\left (\mathcal {O}_{\mathcal {K}_{\sigma }}\right ) \to \mathfrak {IW}^{+}_{w,\mathfrak {c}}(0)_{\mathcal {O}_{\mathbb {C}_{p}}}$
over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)_{\mathcal {O}_{\mathbb {C}_{p}}}$
as in §3.3.1. The additional structures of the Tate object over
$\mathrm {Spec}\left (\breve {R}_{\sigma }\right )$
define a canonical test object
$$ \begin{align*} \left(\mathrm{Tate}_{\mathfrak{a},\mathfrak{b}}(q),\iota_{\mathfrak{a},\mathfrak{b}},\lambda_{\mathfrak{a},\mathfrak{b}},\psi_{\mathfrak{a},\mathfrak{b}}, u_{\mathfrak{a},\mathfrak{b}},\alpha_{\mathfrak{a},\mathfrak{b}}\right) \end{align*} $$
over
$\mathrm {Spec}\left (\mathcal {O}_{\mathcal {K}_{\sigma }}\right )$
. This corresponds via the moduli interpretation to a map
$$ \begin{align*} \tau_{\mathfrak{a},\mathfrak{b},\mathcal{O}_{\mathcal{K}_{\sigma}}}: \mathrm{Spf}\left(\mathcal{O}_{\mathcal{K}_{\sigma}}\right)\to \mathfrak{IW}_{w,\mathfrak{c}}^{+}(0)_{\mathcal{O}_{\mathbb{C}_{p}}} \end{align*} $$
satisfying the following property: the composite
$\breve {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}\to \breve {S}_{\mathscr {C},\mathcal {O}_{\mathbb {C}_{p}}}\overset {\tau _{\mathfrak {a},\mathfrak {b}}}{\to }\mathfrak {IW}_{w,\mathfrak {c}}^{+}(0)_{\mathcal {O}_{\mathbb {C}_{p}}}$
factors through
$\tilde {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
and its restriction to
$\mathrm {Spf}\left (\mathcal {O}_{\mathcal {K}_{\sigma }}\right )$
equals
$\tau _{\mathfrak {a},\mathfrak {b},\mathcal {O}_{\mathcal {K}_{\sigma }}}$
, as in the diagram
Let
$\kappa \in \mathcal {W}\left (\mathbb {C}_{p}\right )$
be any n-analytic weight. Since the formal scheme
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v)_{\mathcal {O}_{\mathbb {C}_{p}}}$
is quasi-compact and the sheaf
$\mathbf {\Omega }^{\kappa }$
is coherent, we have
$$ \begin{align*} M(\mu_{N},\mathfrak{c},\kappa)(v)=H^{0}\left(\bar{\mathfrak{M}}(\mu_{N},\mathfrak{c})(v)_{\mathcal{O}_{\mathbb{C}_{p}}}, \mathbf{\Omega}^{\kappa}\right)[1/p]\subseteq \mathcal{O}\left(\mathfrak{IW}_{w,\mathfrak{c}}^{+}(v)_{\mathcal{O}_{\mathbb{C}_{p}}}\right)[1/p]. \end{align*} $$
For any element
$f_{\mathfrak {c}}$
of
$M(\mu _{N},\mathfrak {c},\kappa )(v)$
, we define the q-expansion
$f_{\mathfrak {c}}(q)$
of
$f_{\mathfrak {c}}$
by
$$ \begin{align*} f_{\mathfrak{c}}(q)=\tau^{*}_{\mathfrak{o},\mathfrak{c}^{-1}}(f_{\mathfrak{c}})\in \mathcal{O}\left(\breve{S}_{\mathscr{C},\mathcal{O}_{\mathbb{C}_{p}}}\right)[1/p]=\mathcal{O}_{\mathbb{C}_{p}}\left[\left[q^{\xi}\mid \xi\in \left(\mathfrak{c}^{-1}\right)^{+}\cup\{0\}\right]\right][1/p]. \end{align*} $$
Thus, for any
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
, we can write
$$ \begin{align*} f_{\mathfrak{c}}(q)=a_{\mathfrak{o},\mathfrak{c}^{-1}}(f,0)+\sum_{\xi\in \left(\mathfrak{c}^{-1}\right)^{+}} a_{\mathfrak{o},\mathfrak{c}^{-1}}(f,\xi)q^{\xi} \end{align*} $$
with some
$a_{\mathfrak {o},\mathfrak {c}^{-1}}(f,\xi )\in \mathbb {C}_{p}$
. For any refinement
$\mathscr {C}^{\prime }\in \mathrm {Dec}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )$
of
$\mathscr {C}$
, the natural map
$\breve {S}_{\mathscr {C}^{\prime },\mathcal {O}_{\mathbb {C}_{p}}}\to \breve {S}_{\mathscr {C},\mathcal {O}_{\mathbb {C}_{p}}}$
induces the identity map on the ring
$\mathcal {O}_{\mathbb {C}_{p}}\left [\left [q^{\xi }\mid \xi \in \left (\mathfrak {c}^{-1}\right )^{+}\cup \{0\}\right ]\right ][1/p]$
. Thus we can compute the q-expansion by taking any refinement of the fixed cone decomposition
$\mathscr {C}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )$
in
$\mathrm {Dec}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )$
. We say that an eigenform f is normalized if
$a_{\mathfrak {o},\mathfrak {o}}(f,1)=1$
.
By presentations (4.2) and (4.5), we have an isomorphism
$$ \begin{align*} \left(\breve{R}_{\sigma}/\hat{I}_{\sigma} \breve{R}_{\sigma}\right)\hat{\otimes}_{\mathcal{O}_{K}}\mathcal{O}_{\mathbb{C}_{p}}\simeq \breve{R}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}/\hat{I}_{\sigma} \breve{R}_{\sigma, \mathcal{O}_{\mathbb{C}_{p}}}, \end{align*} $$
which implies
$a_{\mathfrak {o},\mathfrak {c}^{-1}}(f,0)=0$
if
$f_{\mathfrak {c}}\in S(\mu _{N},\mathfrak {c},\kappa )(v)$
.
4.2 Weak multiplicity one theorem
Let
$(\nu ,w)\in \mathcal {W}^{G}\left (\mathbb {C}_{p}\right )$
be an n-analytic weight. Let
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
be a nonzero eigenform in
$S^{G}(\mu _{N}, (\nu ,w))(v)$
. For any nonzero ideal
$\mathfrak {n}\subseteq \mathfrak {o}$
, let
$\Lambda (\mathfrak {n})$
be the eigenvalue of
$T_{\mathfrak {n}}$
acting on f. We set
$\Phi (\mathfrak {n})$
to be the eigenvalue of
$S_{\mathfrak {n}}$
for
$(\mathfrak {n},Np)=1$
and
${\Phi (\mathfrak {n})=}0$
otherwise. We put
$\mu _{\mathfrak {n}}=\left (\mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}\right )[\mathfrak {n}]$
. Any element
$\zeta $
of
$\mu _{\mathfrak {n}}(L)\subseteq \left (\mathcal {D}^{-1}_{F}\otimes \mathbb {G}_{\mathrm {m}}\right )(L)$
with some extension
$L/K$
defines a ring homomorphism
$\zeta : \mathcal {O}\left (\mathcal {D}^{-1}_{F}\otimes \mathbb {G}_{\mathrm {m}}\right )\to L$
. We put
$\zeta ^{\eta }=\zeta (\mathfrak {X}^{\eta })$
for any
$\eta \in \mathfrak {o}$
, which gives a homomorphism
$\mathfrak {o}/\mathfrak {n}\to L^{\times }$
. We fix an element
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
.
4.2.1 q-Expansion and Hecke operators
For any
$\mathscr {C}\in \mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
and any maximal ideal
$\mathfrak {m}$
of
$\mathfrak {o}$
, we can find
$\mathscr {C}^{\prime }\in \mathrm {Dec}\left (\mathfrak {a},\mathfrak {m}^{-1}\mathfrak {b}\right )$
which is a refinement of
$\mathscr {C}$
. For any
$\sigma \in \mathscr {C}$
and
$\tau \in \mathscr {C}^{\prime }$
satisfying
$\sigma \supseteq \tau $
, we have natural maps
$\hat {R}_{\sigma }\to \hat {R}_{\tau }$
,
$\hat {R}_{\sigma }^{0}\to \hat {R}_{\tau }^{0}$
, and
$\breve {R}_{\sigma }\to \breve {R}_{\tau }$
. Consider the case
$\mathfrak {a}=\mathfrak {o}$
. Let
$\zeta $
be an element of
$\mu _{\mathfrak {m}}(K)$
. Fix an isomorphism of
$\mathfrak {o}$
-modules
$$ \begin{align*} \rho: \mathfrak{m}^{-1}\mathfrak{b}/\mathfrak{b}\simeq \mathfrak{o}/\mathfrak{m}. \end{align*} $$
Then we have a natural ring homomorphism
$$ \begin{align*} q\zeta^{\rho}: \hat{R}_{\tau}\to \hat{R}_{\tau},\quad q^{\xi} \mapsto q^{\xi} \zeta^{\rho(\xi)}. \end{align*} $$
We denote by
$\mathrm {Tate}_{\mathfrak {o},\mathfrak {m}^{-1}\mathfrak {b}}(q\zeta ^{\rho })$
the pullback of
$\mathrm {Tate}_{\mathfrak {o},\mathfrak {m}^{-1}\mathfrak {b}}(q)$
by this map.
On the other hand, we have
$\mathrm {Dec}(\mathfrak {a},\mathfrak {b})=\mathrm {Dec}(\mathfrak {a},\eta \mathfrak {b})$
for any unramified cusp
$(\mathfrak {a},\mathfrak {b},\phi )$
and
$\eta \in F^{\times ,+}$
. Thus any
$\sigma \in \mathscr {C}$
gives similar rings to
$\hat {R}_{\sigma }$
,
$\hat {R}_{\sigma }^{0}$
, and
$\breve {R}_{\sigma }$
for the cusp
$(\mathfrak {a},\eta \mathfrak {b}, \phi )$
, which are denoted by
$\hat {R}_{\eta , \sigma }$
,
$\hat {R}_{\eta ,\sigma }^{0}$
, and
$\breve {R}_{\eta ,\sigma }$
, respectively. We have a natural ring homomorphism
$$ \begin{align*} q^{\eta}: \hat{R}_{\sigma} \to \hat{R}_{\eta,\sigma},\quad q^{\xi}\mapsto q^{\xi\eta}. \end{align*} $$
We denote by
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q^{\eta })$
the pullback of
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
by this map.
We will omit entries of test objects
$(A,\iota ,\lambda ,\psi ,u,\alpha )$
for overconvergent Hilbert modular forms if they are clear from the context.
Lemma 4.3. We have an isomorphism of test objects over
$\hat {R}_{\eta ,\sigma }^{0}$
$$ \begin{align*} \left(\mathrm{Tate}_{\mathfrak{o},\eta\mathfrak{c}^{-1}}(q),\lambda_{\mathfrak{o},\eta\mathfrak{c}^{-1}}\right)\simeq \left(\mathrm{Tate}_{\mathfrak{o},\mathfrak{c}^{-1}}(q^{\eta}),\eta\lambda_{\mathfrak{o},\mathfrak{c}^{-1}}\right). \end{align*} $$
Proof. We denote by
$\rvert _{q^{\eta }}$
the pullback along the map
$q^{\eta }$
. Consider the composite
$$ \begin{align*} \mathfrak{c}^{-1} \to \mathcal{D}_{F}^{-1}\otimes \mathbb{G}_{\mathrm{m}}\left(\hat{R}_{\sigma}^{0}\right)\to \mathcal{D}_{F}^{-1}\otimes \mathbb{G}_{\mathrm{m}}\rvert_{q^{\eta}}\left(\hat{R}_{\eta,\sigma}^{0}\right) \end{align*} $$
of the map
$\alpha \mapsto \left (\mathfrak {X}^{\xi }\mapsto q^{\alpha \xi }\ (\xi \in \mathfrak {o})\right )$
and the map
$q^{\eta }$
, which we also denote by
$q^{\eta }$
. We also have a similar map
$q^{\eta }: \eta \mathfrak {c}^{-1}\to \eta \mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}\rvert _{q^{\eta }}\left (\hat {R}_{\eta ,\sigma }^{0}\right )$
. Then the following diagram over
$\hat {R}_{\eta ,\sigma }^{0}$
is commutative:
This yields an isomorphism
$\mathrm {Tate}_{\mathfrak {o},\eta \mathfrak {c}^{-1}}(q)\to \mathrm {Tate}_{\mathfrak {o},\mathfrak {c}^{-1}}(q^{\eta })$
, as in the lemma.
Lemma 4.4. Let
$\mathfrak {m}$
be a maximal ideal of
$\mathfrak {o}$
satisfying
$\mathfrak {m}\nmid pN$
. Let
$\mathfrak {c}$
be an element of
$\left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
. Take any elements
$x,y\in F^{\times ,+,(p)}$
such that
$\mathfrak {c}^{\prime }=x\mathfrak {m} \mathfrak {c}$
and
$\mathfrak {c}^{\prime \prime }=xy^{-1}\mathfrak {m}^{-1}\mathfrak {c}$
are elements of
$\left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
. Fix an isomorphism of
$\mathfrak {o}$
-modules
$\rho :(x\mathfrak {m}\mathfrak {c})^{-1}/(x\mathfrak {c})^{-1}\simeq \mathfrak {o}/\mathfrak {m}$
. Then we have
$$ \begin{align*} (T_{\mathfrak{m}} f)_{\mathfrak{c}}(q)=\frac{\nu(x)}{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})}\left(\frac{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})^{2}\Phi(\mathfrak{m})}{\nu(y)} f_{\mathfrak{c}^{\prime\prime}}\left(q^{xy^{-1}}\right)+\sum_{\zeta\in \mu_{\mathfrak{m}}\left(\bar{\mathbb{Q}}_{p}\right)}f_{\mathfrak{c}^{\prime}}(q^{x}\zeta^{\rho})\right). \end{align*} $$
Proof. For any
$\mathscr {C}\in \mathrm {Dec}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )$
and
$\mathscr {C}^{\prime }\in \mathrm {Dec}\left (\mathfrak {m}, \mathfrak {c}^{-1}\right )$
, we choose
$\mathscr {C}^{\prime \prime }\in \mathrm {Dec}\left (\mathfrak {o}, (\mathfrak {m}\mathfrak {c})^{-1}\right )$
such that
$\mathscr {C}^{\prime \prime }$
is a common refinement of
$\mathscr {C}$
and
$\mathscr {C}^{\prime }$
. For any
$\sigma \in \mathscr {C}$
and
$\sigma ^{\prime }\in \mathscr {C}^{\prime }$
, take
$\tau \in \mathscr {C}^{\prime \prime }$
satisfying
$\tau \subseteq \sigma ,\sigma ^{\prime }$
. By diagram (4.10) and the inclusions
$$ \begin{align*} \breve{R}_{\tau,\mathcal{O}_{\mathbb{C}_{p}}}\supseteq \tilde{R}_{\tau,\mathcal{O}_{\mathbb{C}_{p}}}\subseteq\mathcal{O}_{\mathcal{K}_{\tau}}, \end{align*} $$
it is enough to show the equality of the lemma after pulling back to
$\mathrm {Spf}\left (\mathcal {O}_{\mathcal {K}_{\tau }}\right )$
.
Choose an element
$\xi _{\mathfrak {m}}\in (x\mathfrak {m}\mathfrak {c})^{-1}$
such that the map
$\rho $
sends the image of
$\xi _{\mathfrak {m}}$
to
$1\in \mathfrak {o}/\mathfrak {m}$
. For any
$\zeta \in \mu _{\mathfrak {m}}\left (\bar {\mathbb {Q}}_{p}\right )$
, we define elements Q and
$Q\zeta $
of
$\mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}\left (\hat {R}^{0}_{x^{-1},\tau }\right )$
by
$$ \begin{align*} \mathfrak{X}^{\eta}\mapsto q^{\xi_{\mathfrak{m}}\eta}\qquad \text{and}\qquad \mathfrak{X}^{\eta}\mapsto q^{\xi_{\mathfrak{m}}\eta}\zeta^{\eta}\quad (\eta\in \mathfrak{o}), \end{align*} $$
respectively. Let
$\mathcal {H}_{Q,\zeta }$
be the
$\mathfrak {m}$
-cyclic
$\mathcal {O}_{F}$
-subgroup of the Tate object
$\mathrm {Tate}_{\mathfrak {o},(x\mathfrak {c})^{-1}}(q)\rvert _{\mathcal {O}_{\mathcal {K}_{\tau }}}$
generated by the image of
$Q\zeta $
. Then, over
$\mathrm {Spec}\left (\mathcal {O}_{\mathcal {K}_{\tau }}\right )$
, the
$\mathfrak {m}$
-cyclic
$\mathcal {O}_{F}$
-subgroup schemes of
$\mathrm {Tate}_{\mathfrak {o},(x\mathfrak {c})^{-1}}(q)$
are exactly
$$ \begin{align*} \mu_{\mathfrak{m}},\qquad \mathcal{H}_{Q,\zeta}\quad \left(\zeta\in \mu_{\mathfrak{m}}\left(\bar{\mathbb{Q}}_{p}\right)\right), \end{align*} $$
where the former is the closed subgroup scheme induced by
$\mu _{\mathfrak {m}}\subseteq \mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}$
. Then the pullback of
$(T_{\mathfrak {m}} f)_{\mathfrak {c}}(q)$
is equal to
$$ \begin{align*} \left(L_{x} T_{\mathfrak{m}} f_{\mathfrak{c}^{\prime}}\right)\left(\mathrm{Tate}_{\mathfrak{o},\mathfrak{c}^{-1}}(q),\lambda_{\mathfrak{o},\mathfrak{c}^{-1}}\right)&=\nu(x) \left(T_{\mathfrak{m}} f_{\mathfrak{c}^{\prime}}\right)\left(\mathrm{Tate}_{\mathfrak{o},\mathfrak{c}^{-1}}(q),x^{-1}\lambda_{\mathfrak{o},\mathfrak{c}^{-1}}\right)\\ &=\nu(x) \left(T_{\mathfrak{m}} f_{\mathfrak{c}^{\prime}}\right)\left(\mathrm{Tate}_{\mathfrak{o},(x\mathfrak{c})^{-1}}(q^{x}),\lambda_{\mathfrak{o},(x\mathfrak{c})^{-1}}\right), \end{align*} $$
which equals
$$ \begin{align*} \frac{\nu(x)}{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})}\left( f_{\mathfrak{c}^{\prime}} \left(\mathrm{Tate}_{\mathfrak{o},(x\mathfrak{c})^{-1}}(q^{x})/\mu_{\mathfrak{m}}\right)+\sum_{\zeta\in\mu_{\mathfrak{m}}\left(\bar{\mathbb{Q}}_{p}\right)}f_{\mathfrak{c}^{\prime}}\left(\mathrm{Tate}_{\mathfrak{o},(x\mathfrak{c})^{-1}}(q^{x})/\mathcal{H}_{Q,\zeta}\rvert_{q^{x}}\right) \right). \end{align*} $$
For the first term, we have the exact sequence
For any
$\xi \in (x\mathfrak {c})^{-1}$
, the natural map
$\mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}\to \mathfrak {m}^{-1}\mathcal {D}_{F}^{-1}\otimes \mathbb {G}_{\mathrm {m}}$
sends the
$\hat {R}_{x^{-1},\tau }^{0}$
-valued point
$\left (\mathfrak {X}^{\eta }\mapsto q^{\xi \eta }\ (\eta \in \mathfrak {o})\right )$
to
$\left (\mathfrak {X}^{\eta }\mapsto q^{\xi \eta }\ (\eta \in \mathfrak {m})\right )$
, and this gives an isomorphism
$$ \begin{align*} \mathrm{Tate}_{\mathfrak{o},(x\mathfrak{c})^{-1}}(q)/\mu_{\mathfrak{m}}\simeq \mathrm{Tate}_{\mathfrak{m},(x\mathfrak{c})^{-1}}(q) \end{align*} $$
compatible with natural additional structures. This implies that the evaluation
$f_{\mathfrak {c}^{\prime }} \left (\mathrm {Tate}_{\mathfrak {o},(x\mathfrak {c})^{-1}}(q^{x})/\mu _{\mathfrak {m}}\right )$
equals
$$ \begin{align*} f_{\mathfrak{c}^{\prime}} (\mathrm{Tate}_{\mathfrak{m},(x\mathfrak{c})^{-1}}(q^{x}),&\lambda_{\mathfrak{m},(x\mathfrak{c})^{-1}})\\ &=f_{\mathfrak{c}^{\prime}} \left(\mathfrak{m}^{-1}\otimes_{\mathcal{O}_{F}} \mathrm{Tate}_{\mathfrak{o},\mathfrak{m}(x\mathfrak{c})^{-1}}(q^{x}),\mathfrak{m}^{2} \lambda_{\mathfrak{o},\mathfrak{m}(x\mathfrak{c})^{-1}}\right)\\ &=\frac{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})^{2}}{\nu(y)}\left(L_{y} S_{\mathfrak{m}} f_{\mathfrak{c}^{\prime}}\right)\left(\mathrm{Tate}_{\mathfrak{o},(\mathfrak{c}^{\prime\prime})^{-1}}\left(q^{xy^{-1}}\right),\lambda_{\mathfrak{o},(\mathfrak{c}^{\prime\prime})^{-1}}\right)\\ &=\frac{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})^{2}\Phi(\mathfrak{m})}{\nu(y)} f_{\mathfrak{c}^{\prime\prime}}\left(\mathrm{Tate}_{\mathfrak{o},(\mathfrak{c}^{\prime\prime})^{-1}}\left(q^{xy^{-1}}\right)\right). \end{align*} $$
For the second term, the subgroup
$$ \begin{align*} \left\{\left(\mathfrak{X}^{\eta}\mapsto q^{\xi\eta}\zeta^{\rho(\xi\eta)}\ (\eta\in \mathfrak{o})\right)\mid \xi\in (\mathfrak{c}^{\prime})^{-1}\right\}\subseteq \mathcal{D}_{F}^{-1}\otimes\mathbb{G}_{\mathrm{m}}\left(\hat{R}_{x^{-1},\tau}^{0}\right) \end{align*} $$
is generated by
$Q\zeta $
and the image of the subgroup
$$ \begin{align*} \left\{\left(\mathfrak{X}^{\eta}\mapsto q^{\xi\eta}\ (\eta\in \mathfrak{o})\right)\mid \xi\in (x\mathfrak{c})^{-1}\right\}\subseteq \mathcal{D}_{F}^{-1}\otimes\mathbb{G}_{\mathrm{m}}\left(\hat{R}_{x^{-1},\sigma}^{0}\right) \end{align*} $$
via the natural map
$\hat {R}_{x^{-1},\sigma }^{0}\to \hat {R}_{x^{-1},\tau }^{0}$
. This yields an isomorphism
$$ \begin{align*} \mathrm{Tate}_{\mathfrak{o},(x\mathfrak{c})^{-1}}(q)/\mathcal{H}_{Q,\zeta} \simeq \mathrm{Tate}_{\mathfrak{o},(\mathfrak{c}^{\prime})^{-1}}(q\zeta^{\rho}) \end{align*} $$
compatible with natural additional structures. Hence the lemma follows.
A similar proof also gives the following variant for
$\mathfrak {m}\mid Np$
:
Lemma 4.5.
-
(1) For any maximal ideal
$\mathfrak {m}\mid N$
, take any element
$x\in F^{\times ,+,(p)}$
satisfying
$\mathfrak {c}^{\prime }=x\mathfrak {m} \mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
. Fix an isomorphism of
$\mathfrak {o}$
-modules
$\rho :(x\mathfrak {m}\mathfrak {c})^{-1}/(x\mathfrak {c})^{-1}\simeq \mathfrak {o}/\mathfrak {m}$
. Then we have
$$ \begin{align*} (T_{\mathfrak{m}} f)_{\mathfrak{c}}(q)=\frac{\nu(x)}{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})}\sum_{\zeta\in \mu_{\mathfrak{m}}\left(\bar{\mathbb{Q}}_{p}\right)}f_{\mathfrak{c}^{\prime}}(q^{x}\zeta^{\rho}). \end{align*} $$
-
(2) For any maximal ideal
$\mathfrak {p}\mid p$
, take any element
$x\in F^{\times ,+,(p)}$
satisfying
$\mathfrak {c}^{\prime }=xx_{\mathfrak {p}}^{-1}\mathfrak {p} \mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
. Fix an isomorphism of
$\mathfrak {o}$
-modules
$\rho :\left (xx_{\mathfrak {p}}^{-1}\mathfrak {p}\mathfrak {c}\right )^{-1}/\left (xx_{\mathfrak {p}}^{-1}\mathfrak {c}\right )^{-1}\simeq \mathfrak {o}/\mathfrak {p}$
. Then we have
$$ \begin{align*} (U_{\mathfrak{p}} f)_{\mathfrak{c}}(q)=\frac{\nu(x)}{\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{p})}\sum_{\zeta\in \mu_{\mathfrak{p}}\left(\bar{\mathbb{Q}}_{p}\right)}f_{\mathfrak{c}^{\prime}}\left(q^{xx_{\mathfrak{p}}^{-1}}\zeta^{\rho}\right). \end{align*} $$
4.2.2 q-Expansion and Hecke eigenvalues
For any
$\xi \in F^{\times }$
, we put
$\chi _{p}(\xi )=\prod _{\mathfrak {p}\mid p} x_{\mathfrak {p}}^{v_{\mathfrak {p}}(\xi )}$
. For any nonzero ideal
$\mathfrak {n}\subseteq \mathfrak {o}$
, take
$\eta \in F^{\times ,+}$
satisfying
$\mathfrak {c}=\eta ^{-1}\mathfrak {n}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
and put
$$ \begin{align*} C(\mathfrak{n},f)=\nu\left(\eta^{-1}\chi_{p}(\eta)\right)a_{\mathfrak{o},\mathfrak{c}^{-1}}(f,\eta). \end{align*} $$
By Lemma 4.3, this is independent of the choice of
$\eta $
. Then we have the following variant of [Reference Shimura37, (2.23)] in our setting:
Lemma 4.6. For any nonzero ideal
$\mathfrak {l},\mathfrak {n}$
of
$\mathfrak {o}$
, we have
$$ \begin{align*} C(\mathfrak{n},T_{\mathfrak{l}} f)=\sum_{\mathfrak{l}+\mathfrak{n}\subseteq \mathfrak{a}\subseteq \mathfrak{o}} \mathrm{N}_{F/\mathbb{Q}}(\mathfrak{a})\Phi(\mathfrak{a}) C\left(\mathfrak{a}^{-2}\mathfrak{l}\mathfrak{n},f\right). \end{align*} $$
Proof. We can easily reduce it to the case
$\mathfrak {l}=\mathfrak {m}^{s}$
for some maximal ideal
$\mathfrak {m}$
. Consider the case of
$\mathfrak {m}\nmid Np$
and
$s=1$
. We follow the notation of Lemma 4.4. Since
$x^{-1}\eta \in (x\mathfrak {c})^{-1}$
, we have
$$ \begin{align*} \sum_{\zeta\in \mu_{\mathfrak{m}}\left(\bar{\mathbb{Q}}_{p}\right)}\zeta^{\rho\left(x^{-1}\eta\right)}= \mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m}). \end{align*} $$
Moreover,
$x^{-1}y\eta \in (\mathfrak {c}^{\prime \prime })^{-1}$
if and only if
$\mathfrak {m} \mid \mathfrak {n}$
. Thus Lemma 4.4 implies
$$ \begin{align*} C(\mathfrak{n}, T_{\mathfrak{m}} f)= \begin{cases} \mathrm{N}_{F/\mathbb{Q}}(\mathfrak{m})\Phi(\mathfrak{m}) C\left(\mathfrak{m}^{-1}\mathfrak{n},f\right)+C(\mathfrak{m}\mathfrak{n},f)& (\mathfrak{m} \mid \mathfrak{n})\\ C(\mathfrak{m}\mathfrak{n},f) & (\mathfrak{m} \nmid \mathfrak{n}) \end{cases} \end{align*} $$
and the lemma follows for this case. The case of
$\mathfrak {m}\mid Np$
and
$s=1$
can be shown similarly from Lemma 4.5. For
$s\geq 2$
, using relation (3.8) we can show the lemma by an induction in the same way as the classical case.
Proposition 4.7. For any
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
and any
$\eta \in \left (\mathfrak {c}^{-1}\right )^{+}$
, put
$\mathfrak {n}=\eta \mathfrak {c}\subseteq \mathfrak {o}$
. Then we have
$$ \begin{align*} a_{\mathfrak{o},\mathfrak{c}^{-1}}(f,\eta)=\nu\left(\eta \chi_{p}(\eta)^{-1}\right) \Lambda(\mathfrak{n}) a_{\mathfrak{o},\mathfrak{o}}(f,1). \end{align*} $$
Proof. We have
$a_{\mathfrak {o},\mathfrak {c}^{-1}}(f,\eta )=\nu \left (\eta \chi _{p}(\eta )^{-1}\right ) C(\mathfrak {n},f)$
and
$C(\mathfrak {o}, f)=a_{\mathfrak {o},\mathfrak {o}}(f,1)$
. By Lemma 4.6, we obtain
$$ \begin{align*} \Lambda(\mathfrak{n})a_{\mathfrak{o},\mathfrak{o}}(f,1)&=\Lambda(\mathfrak{n})C(\mathfrak{o},f)=C(\mathfrak{o}, T_{\mathfrak{n}} f)\\ &=C(\mathfrak{n},f)=\nu\left(\eta^{-1}\chi_{p}(\eta)\right) a_{\mathfrak{o},\mathfrak{c}^{-1}}(f,\eta), \end{align*} $$
from which the proposition follows.
4.3 q-Expansion and integrality
First we show the following lemma:
Lemma 4.8. Let
$\mathfrak {X}$
be a quasi-compact separated admissible formal scheme over
$\mathcal {O}_{\mathbb {C}_{p}}$
. Let
$\mathfrak {F}$
be an invertible sheaf on
$\mathfrak {X}$
. We denote by
$\mathfrak {X}_{\bar {\mathbb {F}}_{p}}$
the special fiber of
$\mathfrak {X}$
and by
$\mathfrak {F}_{\bar {\mathbb {F}}_{p}}$
the pullback of
$\mathfrak {F}$
to
$\mathfrak {X}_{\bar {\mathbb {F}}_{p}}$
.
-
(1) Suppose that
$\mathfrak {X}^{\mathrm {rig}}$
is reduced and
$\mathfrak {X}$
is integrally closed in
$\mathfrak {X}^{\mathrm {rig}}$
. Then for any nonzero element
$f\in H^{0}(\mathfrak {X},\mathfrak {F})[1/p]$
, the
$\mathcal {O}_{\mathbb {C}_{p}}$
-submodule of
$\mathbb {C}_{p}$
is principal.
$$ \begin{align*} I=\left\{x\in\mathbb{C}_{p}\mid x f\in H^{0}(\mathfrak{X},\mathfrak{F})\right\} \end{align*} $$
-
(2) Let g be an element of
$H^{0}(\mathfrak {X},\mathfrak {F})$
. Suppose that the image of g by the map is zero. Then there exists
$$ \begin{align*} H^{0}(\mathfrak{X},\mathfrak{F})\to H^{0}\left(\mathfrak{X}_{\bar{\mathbb{F}}_{p}},\mathfrak{F}_{\bar{\mathbb{F}}_{p}}\right) \end{align*} $$
$x\in m_{\mathbb {C}_{p}}$
satisfying
$g\in xH^{0}(\mathfrak {X},\mathfrak {F})$
.
Proof. For the first assertion, take a finite covering
$\mathfrak {X}=\bigcup _{i=1}^{r} \mathfrak {U}_{i}$
by formal affine open subschemes
$\mathfrak {U}_{i}=\mathrm {Spf}(\mathfrak {A}_{i})$
such that
$\mathfrak {F}\rvert _{\mathfrak {U}_{i}}$
is trivial. Since
$\mathfrak {X}$
is separated, the intersection
$\mathfrak {U}_{i,j}=\mathfrak {U}_{i}\cap \mathfrak {U}_{j}$
is also affine. Put
$A_{i}=\mathfrak {A}_{i}[1/p]$
,
$\mathfrak {M}_{i}=\Gamma (\mathfrak {U}_{i},\mathfrak {F})$
, and
$\mathfrak {M}_{i,j}=\Gamma \left (\mathfrak {U}_{i,j},\mathfrak {F}\right )$
. Then we have a commutative diagram
where the rows are exact and the vertical arrows are injective. Put
$I_{i}=\left \{x\in \mathbb {C}_{p}\mid x f\rvert _{\mathfrak {U}_{i}} \in \mathfrak {M}_{i}\right \}$
. Note that
$I_{i}=\mathbb {C}_{p}$
if
$f\rvert _{\mathfrak {U}_{i}}=0$
. Since the diagram implies
$I=\bigcap _{i=1}^{r} I_{i}$
, it is enough to show that
$I_{i}$
is principal if
$f\rvert _{\mathfrak {U}_{i}}\neq 0$
.
By choosing a trivialization, we identify
$\mathfrak {M}_{i}$
with
$\mathfrak {A}_{i}$
and
$f\rvert _{\mathfrak {U}_{i}}\in \mathfrak {M}_{i}[1/p]$
with a nonzero element
$g_{i}\in A_{i}$
. Note that
$A_{i}$
is a reduced
$\mathbb {C}_{p}$
-affinoid algebra. Since
$\mathfrak {A}_{i}$
is an admissible formal
$\mathcal {O}_{\mathbb {C}_{p}}$
-algebra which is integrally closed in
$A_{i}$
, [Reference Bosch, Güntzer and Remmert10, remark after Proposition 6.3.4/1] implies
$A_{i}^{\circ }=\mathfrak {A}_{i}$
. Thus for any
$x \in \mathbb {C}_{p}$
, we have
$$ \begin{align*} x g_{i} \in \mathfrak{A}_{i} \Leftrightarrow \lvert x\rvert \lvert g_{i}\rvert_{\sup}\leq 1, \end{align*} $$
where
$\lvert g_{i}\rvert _{\sup }$
is the supremum norm of
$g_{i}$
on
$\mathrm {Sp}(A_{i})$
. By the maximum modulus principle, there exists a nonzero element
$\delta \in \mathbb {C}_{p}$
satisfying
$\lvert \delta \rvert =\lvert g_{i}\rvert _{\sup }$
. Hence we obtain
$$ \begin{align*} I_{i}=\left\{x\in \mathbb{C}_{p}\mid \lvert x\rvert\leq \lvert\delta\rvert^{-1}\right\}=\delta^{-1}\mathcal{O}_{\mathbb{C}_{p}}, \end{align*} $$
and the first assertion follows.
For the second assertion, consider the covering
$\mathfrak {X}=\bigcup _{i=1}^{r} \mathfrak {U}_{i}$
as before. Since the reduction of
$g\rvert _{\mathfrak {U}_{i}}$
is also zero, we can write
$g\rvert _{\mathfrak {U}_{i}}=x_{i} h_{i}$
with some
$x_{i}\in m_{\mathbb {C}_{p}}$
and
$h_{i}\in \mathfrak {M}_{i}$
. Replacing
$x_{i}$
by a generator x of the ideal
$(x_{1},\dotsc ,x_{r})$
, we may assume
$g\rvert _{\mathfrak {U}_{i}}=x h_{i}$
for any i. Since
$\mathfrak {M}_{i}$
and
$\mathfrak {M}_{i,j}$
are torsion-free
$\mathcal {O}_{\mathbb {C}_{p}}$
-modules, the elements
$h_{i}$
can be glued to define
$h\in H^{0}(\mathfrak {X},\mathfrak {F})$
. Then we obtain
$g=xh$
and the second assertion follows.
Let
$\kappa \in \mathcal {W}\left (\mathbb {C}_{p}\right )$
be any n-analytic weight. Put
$$ \begin{align*} \mathbf{M}(\mu_{N},\mathfrak{c},\kappa)(0):=H^{0}\left(\bar{\mathfrak{M}}(\mu_{N},\mathfrak{c})(0)_{\mathcal{O}_{\mathbb{C}_{p}}},\mathbf{\Omega}^{\kappa}\right)\subseteq M(\mu_{N},\mathfrak{c},\kappa)(0). \end{align*} $$
This is an
$\mathcal {O}_{\mathbb {C}_{p}}$
-lattice of the Banach
$\mathbb {C}_{p}$
-module
$M(\mu _{N},\mathfrak {c},\kappa )(0)$
. Consider the cusp
$\left (\mathfrak {o},\mathfrak {c}^{-1},\mathrm {id}\right )$
, the fixed cone decomposition
$\mathscr {C}=\mathscr {C}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )\in \mathrm {Dec}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )$
, and
$\sigma \in \mathscr {C}$
. By the definition of the q-expansion, every coefficient of the q-expansion of
$f\in \mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)$
is an element of
$\mathcal {O}_{\mathbb {C}_{p}}$
. We also have the following converse, which can be considered as a q-expansion principle for our setting:
Proposition 4.9. Let
$f_{\mathfrak {c}}$
be any element of
$M(\mu _{N},\mathfrak {c},\kappa )(0)$
. If every coefficient of the q-expansion
$f_{\mathfrak {c}}(q)$
is in
$\mathcal {O}_{\mathbb {C}_{p}}$
, then we have
$f\in \mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)$
.
Proof. Put
$\bar {\mathfrak {M}}^{\mathrm {ord}}=\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(0)$
,
$\bar {\mathfrak {M}}(\Gamma _{1}(p^{n}))^{\mathrm {ord}}=\bar {\mathfrak {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})(0)$
, and
$\mathfrak {IW}^{\mathrm {ord}}=\mathfrak {IW}_{w,\mathfrak {c}}^{+}(0)$
. Recall that
$\mathbf {\Omega }^{\kappa }$
is invertible on
$\bar {\mathfrak {M}}^{\mathrm {ord}}$
. We denote the reduction of
$\bar {\mathfrak {M}}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}$
by
$\bar {\mathfrak {M}}^{\mathrm {ord}}_{\bar {\mathbb {F}}_{p}}$
. Consider the commutative diagram
Recall that
$f_{\mathfrak {c}}\in \mathcal {O}\left (\mathfrak {IW}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}\right )[1/p]$
. The assumption on
$f_{\mathfrak {c}}(q)$
implies
$\tau ^{*}_{\mathfrak {o},\mathfrak {c}^{-1}}(f_{\mathfrak {c}}) \in \mathcal {O}\left (\breve {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}\right )$
.
Consider the special fiber
$$ \begin{align*} \bar{\pi}_{w}: \mathfrak{IW}^{\mathrm{ord}}_{\bar{\mathbb{F}}_{p}}\overset{\bar{\gamma}_{n}}{\to} \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}))^{\mathrm{ord}}_{\bar{\mathbb{F}}_{p}} \overset{\bar{h}_{n}}{\to} \bar{\mathfrak{M}}^{\mathrm{ord}}_{\bar{\mathbb{F}}_{p}} \end{align*} $$
of the map
$\pi _{w}$
and the closed immersion
$i:\bar {\mathfrak {M}}^{\mathrm {ord}}_{\bar {\mathbb {F}}_{p}}\to \bar {\mathfrak {M}}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}$
. From the construction of the sheaf
$\mathbf {\Omega }^{\kappa }\rvert _{\bar {\mathfrak {M}}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}}$
as the fixed part of a
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
-equivariant
$\mathcal {O}_{\mathbb {C}_{p}}$
-flat sheaf on a
$\mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
-torsor, we see that the subsheaf
$\mathbf {\Omega }^{\kappa }\rvert _{\bar {\mathfrak {M}}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}}\subseteq (\pi _{w})_{*}\mathcal {O}_{\mathfrak {IW}_{w,\mathfrak {c}}^{+}(0)_{\mathcal {O}_{\mathbb {C}_{p}}}}$
is formal locally a direct summand. Since
$\pi _{w}$
is affine, for any morphism of formal schemes
$f:S\to \bar {\mathfrak {M}}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}$
the composite of natural maps
$$ \begin{align*} f^{*}\left(\mathbf{\Omega}^{\kappa}\rvert_{\bar{\mathfrak{M}}^{\mathrm{ord}}_{\mathcal{O}_{\mathbb{C}_{p}}}}\right)\to f^{*}(\pi_{w})_{*}\mathcal{O}_{\mathfrak{IW}^{\mathrm{ord}}_{\mathcal{O}_{\mathbb{C}_{p}}}}\to \left(\pi_{w}\rvert_{\pi_{w}^{-1}(S)}\right)_{*}\mathcal{O}_{\pi_{w}^{-1}(S)} \end{align*} $$
is injective. This yields a commutative diagram
with injective horizontal arrows, where the base extension
$\hat {S}_{\mathscr {C},\bar {\mathbb {F}}_{p}}=\hat {S}_{\mathscr {C}}\hat {\otimes }_{k}\bar {\mathbb {F}}_{p}$
is equal to the special fiber of
$\breve {S}_{\mathscr {C},\mathcal {O}_{\mathbb {C}_{p}}}$
.
On the Tate object
$\mathrm {Tate}_{\mathfrak {o},\mathfrak {c}^{-1}}(q)$
over
$\mathrm {Spec}\left (\breve {R}_{\sigma }\right )$
, we defined the canonical trivialization of the canonical subgroup and that of the
$\mathbb {T}_{w}^{0}\left (\breve {S}_{\sigma }\right )$
-set
$\mathfrak {IW}^{\mathrm {ord}}\left (\breve {S}_{\sigma }\right )$
, which are denoted by
$u_{\mathfrak {o},\mathfrak {c}^{-1}}$
and
$\alpha _{\mathfrak {o},\mathfrak {c}^{-1}}$
. For any
$a\in \mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
, we fix its lift
$\hat {a}\in \mathbb {T}_{w}\left (\mathbb {Z}_{p}\right )$
satisfying
$\hat {1}=1$
. Since
$\breve {R}_{\sigma }$
is Noetherian, the moduli interpretation of
$\mathfrak {IW}^{\mathrm {ord}}$
is available over
$\breve {R}_{\sigma }$
, and these trivializations give isomorphisms
$$ \begin{align*} \breve{S}_{\sigma}\times_{\bar{\mathfrak{M}}^{\mathrm{ord}}} \bar{\mathfrak{M}}(\Gamma_{1}(p^{n}))^{\mathrm{ord}}\simeq \coprod_{a\in \mathbb{T}\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)} \breve{S}_{\sigma},\qquad \pi_{w}^{-1}\left(\breve{S} {}_{\sigma}\right)\simeq \coprod_{a\in \mathbb{T}\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)} \breve{S}_{\sigma}\times\mathbb{T}_{w}^{0}. \end{align*} $$
The latter is an isomorphism of formal
$\mathbb {T}_{w}^{0}$
-torsors given by
$\left (au_{\mathfrak {o},\mathfrak {c}^{-1}},\hat {a}\alpha _{\mathfrak {o},\mathfrak {c}^{-1}}\right )$
at the a-component. By the base extension, we also have similar isomorphisms over
$\mathcal {O}_{\mathbb {C}_{p}}$
. Since the latter isomorphism is defined by
$\left (u_{\mathfrak {o},\mathfrak {c}^{-1}},\alpha _{\mathfrak {o},\mathfrak {c}^{-1}}\right )$
at the component
$a=1$
, the unit section on this component coincides with the foregoing map
$\breve {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}} \to \pi _{w}^{-1} \left (\breve {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}\right )$
.
For any formal character
$\chi $
of
$\mathbb {T}_{w}^{0}$
, the
$\chi $
-part of the
$\mathbb {T}_{w}^{0}$
-representation
$\mathcal {O}\left (\breve {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}\times \mathbb {T}_{w}^{0}\right )$
is a free
$\breve {R} {}_{\sigma , \mathcal {O}_{\mathbb {C}_{p}}}$
-module of rank
$1$
with a generator
$s_{\chi }$
. Then we have
$$ \begin{align*} f_{\mathfrak{c}}\rvert_{\pi^{-1}_{w}\left(\breve{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\right)}\in \prod_{a\in \mathbb{T}\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)} \left(\breve{R}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}[1/p]s_{\kappa^{-1}}\right). \end{align*} $$
Write this element as
$\left (F_{a} s_{\kappa ^{-1}}\right )_{a\in \mathbb {T}\left (\mathbb {Z}/p^{n}\mathbb {Z}\right )}$
with
$F_{a}\in \breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}[1/p]$
. Since
$\kappa (1)=1$
and
$\tau _{\mathfrak {o},\mathfrak {c}^{-1}} {}^{*}(f_{\mathfrak {c}})\in \breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
, we obtain
$F_{1}\in \breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
. Since
$f_{\mathfrak {c}}$
is
$\kappa ^{-1}$
-equivariant for the
$\mathbb {T}_{w}\left (\mathbb {Z}_{p}\right )$
-action, we have
$F_{a}=\kappa \left (\hat {a}\right )F_{1}$
. Since the image of the character
$\kappa $
is contained in
$\mathcal {O}_{\mathbb {C}_{p}}^{\times }$
, we see that
$F_{a}\in \breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
for any
$a\in \mathbb {T}(\mathbb {Z}/p^{n}\mathbb {Z})$
. This means
$$ \begin{align} f_{\mathfrak{c}}\rvert_{\pi^{-1}_{w}\left(\breve{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\right)}\in \mathcal{O}\left(\pi_{w}^{-1}\left(\breve{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\right)\right). \end{align} $$
To prove the proposition, we may assume
$f_{\mathfrak {c}}\neq 0$
. Consider the ideal
$J=\left \{x\in \mathcal {O}_{\mathbb {C}_{p}} \mid x f_{\mathfrak {c}} \in \mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)\right \}$
, which is principal by Lemma 4.8(1). Put
$J=(x)$
and suppose
$x\in m_{\mathbb {C}_{p}}$
. Then the q-expansion
$x f_{\mathfrak {c}}(q)$
is also integral, and zero modulo
$m_{\mathbb {C}_{p}}$
. Thus the commutative diagram (4.11) and formula (4.12) imply that the pullback of
$x f_{\mathfrak {c}}\in \mathbf {\Omega }^{\kappa }\left (\bar {\mathfrak {M}}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}\right )$
to
$i^{*}\mathbf {\Omega }^{\kappa }\rvert _{\hat {S}_{\mathscr {C},\bar {\mathbb {F}}_{p}}}\left (\hat {S}_{\mathscr {C},\bar {\mathbb {F}}_{p}}\right )$
vanishes.
Note that the reduction of
$\breve {S}_{\mathscr {C},\mathcal {O}_{\mathbb {C}_{p}}}\to \bar {\mathfrak {M}}^{\mathrm {ord}}_{\mathcal {O}_{\mathbb {C}_{p}}}$
induces the map on the special fiber
$$ \begin{align*} \hat{S}_{\mathscr{C},\bar{\mathbb{F}}_{p}}=\left(\breve{S}_{\mathscr{C},\mathcal{O}_{\mathbb{C}_{p}}}\right)_{\bar{\mathbb{F}}_{p}} \to \bar{M}(\mu_{N},\mathfrak{c})_{\bar{\mathbb{F}}_{p}}. \end{align*} $$
Let
$\bar {M}(\mu _{N},\mathfrak {c})_{\bar {\mathbb {F}}_{p}}\rvert ^{\wedge }_{D_{\bar {\mathbb {F}}_{p}}}$
be the formal completion of
$\bar {M}(\mu _{N},\mathfrak {c})_{\bar {\mathbb {F}}_{p}}$
along its boundary
$D_{\bar {\mathbb {F}}_{p}}$
. Recall that this map induces maps
$$ \begin{align*} \hat{S}_{\mathscr{C},\bar{\mathbb{F}}_{p}} \to \hat{S}_{\mathscr{C},\bar{\mathbb{F}}_{p}}/U_{N} \to \bar{M}(\mu_{N},\mathfrak{c})_{\bar{\mathbb{F}}_{p}}\rvert^{\wedge}_{D_{\bar{\mathbb{F}}_{p}}}, \end{align*} $$
where the first arrow is a surjective local isomorphism and the second arrow is an open immersion. Hence
$x f_{\mathfrak {c}}$
vanishes on a formal open subscheme of the formal completion
$\bar {M}(\mu _{N},\mathfrak {c})_{\bar {\mathbb {F}}_{p}}\rvert ^{\wedge }_{D_{\bar {\mathbb {F}}_{p}}}$
. We know that the smooth scheme
$\bar {\mathfrak {M}}^{\mathrm {ord}}_{\bar {\mathbb {F}}_{p}}$
is irreducible. Since the sheaf
$\mathbf {\Omega }^{\kappa }$
is invertible on the ordinary locus, Krull’s intersection theorem implies that
$x f_{\mathfrak {c}}$
vanishes on a nonempty open subscheme of
$\bar {\mathfrak {M}}^{\mathrm {ord}}_{\bar {\mathbb {F}}_{p}}$
, and thus it also vanishes on
$\bar {\mathfrak {M}}^{\mathrm {ord}}_{\bar {\mathbb {F}}_{p}}$
. Then Lemma 4.8(2) implies that
$x f_{\mathfrak {c}}\in y \mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)$
for some
$y\in m_{\mathbb {C}_{p}}$
. Since the
$\mathcal {O}_{\mathbb {C}_{p}}$
-module
$\mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)$
is torsion free, this contradicts the choice of x. Thus we obtain
$x\in \mathcal {O}_{\mathbb {C}_{p}}^{\times }$
and
$f_{\mathfrak {c}} \in \mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)$
, which concludes the proof of the proposition.
Corollary 4.10. Let
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}}$
be a nonzero eigenform in the space
$S^{G}(\mu _{N},(\nu ,w))(v)$
of weight
$(\nu ,w)\in \mathcal {W}^{G}\left (\mathbb {C}_{p}\right )$
. For any nonzero ideal
$\mathfrak {n}$
of
$\mathfrak {o}$
, the Hecke eigenvalue
$\Lambda (\mathfrak {n})$
is p-integral.
Proof. By equation (3.8), it is enough to show the case where
$\mathfrak {n}$
is a maximal ideal
$\mathfrak {m}$
. Put
$\kappa =k(\nu ,w)$
. Note that by Lemmas 3.1 and 3.4, the restriction map
$S^{G}(\mu _{N},(\nu ,w))(v)\to S^{G}(\mu _{N},(\nu ,w))(0)$
is injective. We consider
$\Lambda (\mathfrak {m})$
as an eigenvalue of the operator
$T_{\mathfrak {m}}$
acting on
$$ \begin{align*} M:=\bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} M(\mu_{N},\mathfrak{c},\kappa)(0). \end{align*} $$
This is a Banach
$\mathbb {C}_{p}$
-module with respect to the p-adic norm
$\lvert -\rvert $
defined by the
$\mathcal {O}_{\mathbb {C}_{p}}$
-lattice
$$ \begin{align*} \mathbf{M}:=\bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathbf{M}(\mu_{N},\mathfrak{c},\kappa)(0). \end{align*} $$
Namely, we put
$$ \begin{align*} \lvert f\rvert=\inf\left\{\lvert x\rvert^{-1}\mid x\in \mathbb{C}_{p}^{\times},\ xf\in \mathbf{M}\right\}. \end{align*} $$
By Lemma 4.8(1), we can find an element
$x\in \mathbb {C}_{p}$
of largest absolute value satisfying
$x f_{\mathfrak {c}}\in \mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)$
for any
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
. The norm
$\lvert f\rvert $
is equal to
$\lvert x\rvert ^{-1}$
. Moreover, any coefficient of the q-expansion
$x f_{\mathfrak {c}}(q)$
is contained in
$\mathcal {O}_{\mathbb {C}_{p}}$
. By Lemma 4.6, so is
$x T_{\mathfrak {m}} f$
. Hence Proposition 4.9 shows
$x T_{\mathfrak {m}} f\in \mathbf {M}$
. This implies
$$ \begin{align*} \lvert\Lambda(\mathfrak{m})\rvert=\frac{\lvert T_{\mathfrak{m}} f\rvert}{\lvert f\rvert}\leq \frac{\lvert x\rvert^{-1}}{\lvert x\rvert^{-1}}=1, \end{align*} $$
and the corollary follows.
Corollary 4.11. Let
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
be a normalized eigenform in
$S^{G}(\mu _{N},(\nu ,w))(v)$
of weight
$(\nu ,w)\in \mathcal {W}^{G}\left (\mathbb {C}_{p}\right )$
. Then we have
$$ \begin{align*} a_{\mathfrak{o},\mathfrak{c}^{-1}}(f,\eta)\in \mathcal{O}_{\mathbb{C}_{p}} \end{align*} $$
for any
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
and any
$\eta \in \left (\mathfrak {c}^{-1}\right )^{+}$
.
Corollary 4.12. Let
$(\nu ,w)$
be an element of
$\mathcal {W}^{G}\left (\mathbb {C}_{p}\right )$
.
-
(1) For any
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
, there exists an admissible affinoid open subset
$\mathcal {S}_{\mathfrak {c}}\subseteq \bar {\mathcal {M}}(\mu _{N}, \mathfrak {c})(v)_{\mathbb {C}_{p}}$
such that
$\left (\pi _{w}^{\mathrm {rig}}\right )^{-1}(\mathcal {S}_{\mathfrak {c}})$
meets every connected component of
$\mathcal {IW}^{+}_{w,\mathfrak {c}}(v)_{\mathbb {C}_{p}}$
and, for any normalized eigenform
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
in
$S^{G}(\mu _{N},(\nu ,w))(v)$
, the restriction
$f_{\mathfrak {c}}\rvert _{\left (\pi _{w}^{\mathrm {rig}}\right )^{-1}\left (\mathcal {S}_{\mathfrak {c}}\right )}$
has absolute value bounded by
$1$
. -
(2) Let
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
be any element in the space
$S^{G}(\mu _{N},(\nu ,w))(v)$
. If
$f_{\mathfrak {c}}(q)=0$
for any
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
, then
$f=0$
. -
(3) Let
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
and
$f^{\prime }=\left (f^{\prime }_{\mathfrak {c}}\right )_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
be normalized eigenforms in
$S^{G}(\mu _{N},(\nu ,w))(v)$
. Suppose that the eigenvalues of the Hecke operator
$T_{\mathfrak {n}}$
acting on f and
$f^{\prime }$
are the same for any nonzero ideal
$\mathfrak {n}\subseteq \mathfrak {o}$
. Then
$f=f^{\prime }$
.
Proof. Let us prove the first assertion. For any
$\sigma \in \mathscr {C}=\mathscr {C}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )$
, Corollary 4.11 and formula (4.7) show that
$\tau _{\mathfrak {o}, \mathfrak {c}^{-1}}^{*}(f_{\mathfrak {c}})$
is a rigid analytic function on
$\breve {S}^{\mathrm {rig}}_{\sigma ,\mathbb {C}_{p}}$
with absolute value bounded by
$1$
. As in the proof of Proposition 4.9, we can show that
$f_{\mathfrak {c}}\rvert _{\left (\pi _{w}^{\mathrm {rig}}\right )^{-1}\left (\breve {S}^{\mathrm {rig}}_{\mathscr {C},\mathbb {C}_{p}}\right )}$
is a rigid analytic function with absolute value bounded by
$1$
. Since the natural map
$\breve {S}^{\mathrm {rig}}_{\mathscr {C},\mathbb {C}_{p}} \to \breve {S}_{\mathscr {C},\mathbb {C}_{p}}^{\mathrm {rig}}/U_{N}$
is a surjective local isomorphism, the restriction
$f_{\mathfrak {c}}\rvert _{\left (\pi _{w}^{\mathrm {rig}}\right )^{-1} \left (\breve {S}_{\mathscr {C},\mathbb {C}_{p}}^{\mathrm {rig}}/U_{N}\right )}$
is also, with absolute value bounded by
$1$
. Thus, for any nonempty admissible affinoid open subset
$\mathcal {S}_{\mathfrak {c}}\subseteq \breve {S}_{\mathscr {C},\mathbb {C}_{p}}^{\mathrm {rig}}/U_{N}$
, the absolute value of
$f_{\mathfrak {c}}\rvert _{\left (\pi _{w}^{\mathrm {rig}}\right )^{-1}\left (\mathcal {S}_{\mathfrak {c}}\right )}$
is bounded by
$1$
. Since
$\breve {S}_{\mathscr {C},\mathbb {C}_{p}}^{\mathrm {rig}}/U_{N}$
is an admissible open subset of
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)_{\mathbb {C}_{p}}$
, we see that
$\mathcal {S}_{\mathfrak {c}}$
is also its admissible open subset.
On the other hand, the rigid analytic variety
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)_{\mathbb {C}_{p}}$
is connected by Lemma 3.4. Since the map
$$ \begin{align*} h^{\mathrm{rig}}_{n}: \bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v)_{\mathbb{C}_{p}}\to \bar{\mathcal{M}}(\mu_{N},\mathfrak{c})(v)_{\mathbb{C}_{p}} \end{align*} $$
is finite and étale [Reference Conrad18, Theorem A.2.4], it is surjective on each connected component of the rigid analytic variety
$\bar {\mathcal {M}}(\Gamma _{1}(p^{n}),\mu _{N}, \mathfrak {c})(v)_{\mathbb {C}_{p}}$
, and thus
$\left (h_{n}^{\mathrm {rig}}\right )^{-1}(\mathcal {S}_{\mathfrak {c}})$
meets every connected component of it.
We claim that the map
$$ \begin{align*} \gamma_{w}^{\mathrm{rig}}: \mathcal{IW}_{w,\mathfrak{c}}^{+}(v)_{\mathbb{C}_{p}}\to \bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v)_{\mathbb{C}_{p}} \end{align*} $$
induces a bijection
$$ \begin{align*} \pi_{0}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v)_{\mathbb{C}_{p}}\right)\to \pi_{0}\left(\bar{\mathcal{M}}(\Gamma_{1}(p^{n}),\mu_{N},\mathfrak{c})(v)_{\mathbb{C}_{p}}\right) \end{align*} $$
between the sets of connected components. Indeed, by [Reference Conrad17, Corollary 3.2.3], it is enough to show the claim with
$\mathbb {C}_{p}$
replaced by a finite extension
$L/K$
. By a finite base extension, we may assume
$L=K$
. Since the formal schemes
$ \mathfrak {IW}_{w,\mathfrak {c}}^{+}(v)$
and
$\bar {\mathfrak {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})(v)$
are both normal, it is enough to show a similar assertion for the formal model
$\gamma _{w}$
. Since it is a formal
$\mathbb {T}^{0}_{w}$
-torsor, it is surjective and the map between the sets of connected components is also surjective. Let
$\mathfrak {Y}$
be any connected component of
$\bar {\mathfrak {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})(v)$
and let
$\left \{\mathfrak {X}_{j}\right \}_{j\in J}$
be the set of connected components of
$\mathfrak {IW}_{w,\mathfrak {c}}^{+}(v)$
which
$ \gamma _{w}$
maps to
$\mathfrak {Y}$
. Suppose
$\sharp J\geq 2$
. Since
$\gamma _{w}$
is finitely presented and flat, it is open and the connectedness of
$\mathfrak {Y}$
implies that
$ \gamma _{w}\left (\mathfrak {X}_{j}\right )\cap \gamma _{w}\left (\mathfrak {X}_{j^{\prime }}\right )\neq \emptyset $
for some
$j\neq j^{\prime }$
. However, for any element y of this intersection, the fiber
$\gamma _{w}^{-1}(y)$
is connected, since it is isomorphic to the special fiber of
$\mathbb {T}^{0}_{w}$
, which is a contradiction. Since
$\gamma _{w}^{\mathrm {rig}}$
is surjective, the claim shows that every connected component of
$\mathcal {IW}^{+}_{w,\mathfrak {c}}(v)_{\mathbb {C}_{p}}$
meets the admissible open subset
$\left (\pi _{w}^{\mathrm {rig}}\right )^{-1}(\mathcal {S}_{\mathfrak {c}})$
, and the first assertion follows.
Now suppose that
$f_{\mathfrak {c}}(q)=0$
for any
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
. Then we have
$f_{\mathfrak {c}}\rvert _{\left (\pi _{w}^{\mathrm {rig}}\right )^{-1}\left (\mathcal {S}_{\mathfrak {c}}\right )}=0$
. Since the rigid analytic variety
$\mathcal {IW}^{+}_{w,\mathfrak {c}}(v)_{\mathbb {C}_{p}}$
is smooth over
$\mathbb {C}_{p}$
, the first assertion and Lemma 3.1 show the second assertion. The third assertion follows from Proposition 4.7 and the second one.
4.4 Normalized overconvergent modular forms in families
Let
$\mathcal {U}=\mathrm {Sp}(A)$
be a smooth
$\mathbb {C}_{p}$
-affinoid variety and put
$\mathfrak {U}=\mathrm {Spf}(A^{\circ })$
. Let
$\mathcal {U}\to \mathcal {W}^{G}_{\mathbb {C}_{p}}$
be an n-analytic morphism and consider the associated weight characters
$\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )$
as before. Let
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
be an eigenform in the space
$S^{G}\left (\mu _{N}, \left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v)$
. Recall that each
$f_{\mathfrak {c}}$
is an element of
$\mathcal {O}\left (\mathfrak {IW}_{w,\mathfrak {c}}^{+}(v)_{\mathcal {O}_{\mathbb {C}_{p}}}\times \mathfrak {U}\right )[1/p]$
. For the cusp
$\left (\mathfrak {o},\mathfrak {c}^{-1},\mathrm {id}\right )$
of
$M(\mu _{N},\mathfrak {c})$
and any
$\sigma \in \mathscr {C}=\mathscr {C}\left (\mathfrak {o},\mathfrak {c}^{-1}\right )$
, we have the map
$$ \begin{align*} \tau_{\mathfrak{o},\mathfrak{c}^{-1}}\times 1: \breve{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}} \times \mathfrak{U} \to \mathfrak{IW}_{w,\mathfrak{c}}^{+}(v)_{\mathcal{O}_{\mathbb{C}_{p}}}\times \mathfrak{U} \end{align*} $$
over
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v)_{\mathcal {O}_{\mathbb {C}_{p}}}\times \mathfrak {U}$
.
As in §4.1, we see that the ring
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}\hat {\otimes }_{\mathcal {O}_{\mathbb {C}_{p}}}A^{\circ }$
is isomorphic to the completion of the ring
$$ \begin{align*} A^{\circ}\left[q^{\xi_{1}},\dotsc, q^{\xi_{r}}\right]\left[q^{\pm \xi_{r+1}},\dotsc, q^{\pm \xi_{g}}\right] \end{align*} $$
with respect to the
$\left (p, q^{\xi _{1}}\dotsm q^{\xi _{r}}\right )$
-adic topology for some
$\xi _{1},\dotsc ,\xi _{g}\in \mathfrak {c}^{-1}\cap \sigma ^{\vee }$
, and thus it can be considered as a subring of the ring
$$ \begin{align*} A^{\circ}\left\langle q^{\pm \xi_{r+1}},\dotsc, q^{\pm \xi_{g}}\right\rangle \left[\left[q^{\xi_{1}},\dotsc, q^{\xi_{r}}\right]\right]. \end{align*} $$
Hence we obtain the map of the
$q^{1}$
-coefficient
$$ \begin{align*} \mathrm{pr}_{q^{1}}^{\mathcal{U}}: \mathcal{O}\left(\breve{S}_{\sigma,\mathcal{O}_{\mathbb{C}_{p}}}\times \mathfrak{U}\right)[1/p]\to A. \end{align*} $$
For any eigenform
$f\in S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v)$
as before, we put
$a_{\mathfrak {o},\mathfrak {o}}^{\mathcal {U}}(f,1)=\mathrm {pr}_{q^{1}}^{\mathcal {U}}((\tau _{\mathfrak {o},\mathfrak {o}}\times 1)^{*}(f_{\mathfrak {o}}))\in A$
.
For any
$x\in \mathcal {U}\left (\mathbb {C}_{p}\right )$
, put
$(\nu ,w)=\left (\nu ^{\mathcal {U}}(x),w^{\mathcal {U}}(x)\right )$
. The specialization
$f(x)=(f_{\mathfrak {c}}(x))_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
is an element of the space
$S^{G}(\mu _{N},(\nu ,w))(v)$
over
$\mathbb {C}_{p}$
, and we have the usual
$q^{1}$
-coefficient
$a_{\mathfrak {o},\mathfrak {o}}(f(x),1)$
of the q-expansion of
$f(x)$
. By the commutative diagram
we obtain
$$ \begin{align} a_{\mathfrak{o},\mathfrak{o}}^{\mathcal{U}}(f,1)(x)=a_{\mathfrak{o},\mathfrak{o}}(f(x),1). \end{align} $$
Lemma 4.13. Suppose that
$f(x)\neq 0$
for any
$x\in \mathcal {U}\left (\mathbb {C}_{p}\right )$
. Then we have
$$ \begin{align*} a_{\mathfrak{o},\mathfrak{o}}^{\mathcal{U}}(f,1)\in A^{\times}. \end{align*} $$
In particular, the specialization
$f^{\prime }(x)$
of
$f^{\prime }=a_{\mathfrak {o},\mathfrak {o}}^{\mathcal {U}}(f,1)^{-1} f$
is a normalized eigenform with the same eigenvalues as
$f(x)$
for any
$x\in \mathcal {U}\left (\mathbb {C}_{p}\right )$
.
Proof. We claim that
$a_{\mathfrak {o},\mathfrak {o}}(f(x),1)\neq 0$
for any
$x\in \mathcal {U}\left (\mathbb {C}_{p}\right )$
. Indeed, suppose that
$a_{\mathfrak {o},\mathfrak {o}}(f(x),1)=0$
for some
$x\in \mathcal {U}\left (\mathbb {C}_{p}\right )$
. Since
$f(x)$
is an eigenform, Proposition 4.7 implies that the q-expansion
$f(x)_{\mathfrak {c}}(q)$
of
$f(x)$
is zero for any
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
. By Corollary 4.12(2) we have
$f(x)=0$
, which is a contradiction.
Now equation (4.13) implies that
$a_{\mathfrak {o},\mathfrak {o}}^{\mathcal {U}}(f,1)(x)\neq 0$
for any
$x\in \mathcal {U}\left (\mathbb {C}_{p}\right )$
. Hence we obtain
$a_{\mathfrak {o},\mathfrak {o}}^{\mathcal {U}}(f,1)\in A^{\times }$
.
4.5 Gluing results
Here we prove two results on gluing overconvergent Hilbert modular forms, based on the theory of the q-expansion developed in the foregoing. Let
$\mathcal {X}=\mathrm {Sp}(R)$
be any admissible affinoid open subset of
$\mathcal {W}^{G}$
. Put
$n=n(\mathcal {X})$
and
$v=v_{n}$
as in §3.3.3. Consider the Hilbert eigenvariety
$\mathcal {E}\rvert _{\mathcal {X}}\to \mathcal {X}$
, which is constructed from the input data
$$ \begin{align*} \left(R,S^{G}\left(\mu_{N},\left(\nu^{\mathcal{X}},w^{\mathcal{X}}\right)\right)(v_{\mathrm{tot}}),\mathbf{T},U_{p}\right). \end{align*} $$
4.5.1 Gluing local eigenforms
Lemma 4.14. Let
$\mathcal {U}=\mathrm {Sp}(A)$
be a smooth
$\mathbb {C}_{p}$
-affinoid variety and let
$\mathcal {U}\to \mathcal {X}_{\mathbb {C}_{p}}$
be a morphism of rigid analytic varieties over
$\mathbb {C}_{p}$
. Let f be an eigenvector of the space
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {X}},w^{\mathcal {X}}\right )\right )(v_{\mathrm {tot}})\hat {\otimes }_{R} A$
for the action of
$\mathbf {T}$
such that for any
$x\in \mathcal {U}\left (\mathbb {C}_{p}\right )$
, the specialization
$$ \begin{align*} f(x)\in S^{G}\left(\mu_{N},\left(\nu^{\mathcal{X}},w^{\mathcal{X}}\right)\right)(v_{\mathrm{tot}})\hat{\otimes}_{R,x^{*}} \mathbb{C}_{p} \end{align*} $$
is nonzero. Then the image of f by the natural map
$$ \begin{align*} S^{G}\left(\mu_{N},\left(\nu^{\mathcal{X}},w^{\mathcal{X}}\right)\right)(v_{\mathrm{tot}})\hat{\otimes}_{R} A\to S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}}) \end{align*} $$
is an eigenform with the same property.
Proof. Put
$(\nu ,w)=\left (\nu ^{\mathcal {U}}(x),w^{\mathcal {U}}(x)\right )$
. Then we have the commutative diagram
Here the lowest two arrows are the specialization maps. Since
$\mathcal {W}^{G}$
is smooth, the maximal ideal of
$R\hat {\otimes }_{K}\mathbb {C}_{p}$
corresponding to x is generated by a regular sequence. By Lemma 3.13, the left oblique arrow is an isomorphism. This implies the lemma.
Proposition 4.15. Let
$\mathcal {Z}$
be a smooth rigid analytic variety over
$\mathbb {C}_{p}$
which is principally refined. Let
$\varphi :\mathcal {Z}\to (\mathcal {E}\rvert _{\mathcal {X}})_{\mathbb {C}_{p}}$
be a morphism of rigid analytic varieties over
$\mathbb {C}_{p}$
. Then there exist an element
$$ \begin{align*} f\in \bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v_{\mathrm{tot}})_{\mathbb{C}_{p}}\times \mathcal{Z}\right) \end{align*} $$
and an admissible affinoid covering
$\mathcal {Z}=\bigcup _{i\in I} \mathcal {U}_{i}$
such that the restriction
$f\rvert _{\mathcal {U}_{i}}$
for each
$i\in I$
is an eigenform of
$S^{G}\left (\mu _{N}, \left (\nu ^{\mathcal {U}_{i}},w^{\mathcal {U}_{i}}\right )\right )(v_{\mathrm {tot}})$
with eigensystem
$\varphi ^{*}: \mathbf {T}\to \mathcal {O}(\mathcal {Z})\to \mathcal {O}(\mathcal {U}_{i})$
and
$f(z)$
is normalized for any
$z\in \mathcal {Z}$
.
Proof. By Proposition 2.5(2), there exist an admissible affinoid covering
$\mathcal {Z}=\bigcup _{i\in I} \mathcal {U}_{i}$
,
$\mathcal {U}_{i}=\mathrm {Sp}(A_{i})$
with a principal ideal domain
$A_{i}$
, and an eigenvector
$f_{i}$
in the space
$$ \begin{align*} S^{G}\left(\mu_{N},\left(\nu^{\mathcal{X}},w^{\mathcal{X}}\right)\right)(v_{\mathrm{tot}})\hat{\otimes}_{R} A_{i} \end{align*} $$
such that for any
$z\in \mathcal {U}_{i}$
, we have
$f_{i}(z)\neq 0$
and
$$ \begin{align*} (h\otimes 1)f_{i}=(1\otimes \varphi^{*}(h))f_{i} \end{align*} $$
for any
$h\in \mathbf {T}$
. By Lemma 4.14, the image
$f^{\prime }_{i}$
of
$f_{i}$
in the space
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}_{i}},w^{\mathcal {U}_{i}}\right )\right )(v_{\mathrm {tot}})$
is an eigenform with eigensystem
$\varphi ^{*}:\mathbf {T}\to A_{i}$
such that
$f^{\prime }_{i}(z)\neq 0$
for any
$z\in \mathcal {U}_{i}$
. Since
$\mathcal {U}_{i}$
is smooth, by Lemma 4.13 we may assume that
$f^{\prime }_{i}(z)$
is a normalized eigenform for any
$z\in \mathcal {U}_{i}$
. For any
$z\in \mathcal {U}_{i}\cap \mathcal {U}_{j}$
and any
$h\in \mathbf {T}$
, the h-eigenvalues of
$f^{\prime }_{i}(z)$
and
$f^{\prime }_{j}(z)$
are both
$\varphi ^{*}(h)(z)$
. Since they are normalized eigenforms, Corollary 4.12(3) implies that the images of
$f^{\prime }_{i}(z)$
and
$f^{\prime }_{j}(z)$
in
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}_{i}}(z),w^{\mathcal {U}_{i}}(z)\right )\right ) (0)$
agree with each other. By Lemmas 3.1 and 3.4, we obtain
$f^{\prime }_{i}(z)=f^{\prime }_{j}(z)$
.
Since the rigid analytic variety
$\mathcal {IW}^{+}_{w,\mathfrak {c}}(v_{\mathrm {tot}})_{\mathbb {C}_{p}}\times \mathcal {Z}$
is reduced, this equality means that
$f^{\prime }_{i}$
and
$f^{\prime }_{j}$
coincide with each other as rigid analytic functions on
$$ \begin{align*} \coprod_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathcal{IW}^{+}_{w,\mathfrak{c}}(v_{\mathrm{tot}})_{\mathbb{C}_{p}} \times \left(\mathcal{U}_{i}\cap\mathcal{U}_{j}\right). \end{align*} $$
Thus we can glue
$f^{\prime }_{i}$
s to produce an element
$$ \begin{align*} f\in \bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}}\mathcal{O}\left(\mathcal{IW}^{+}_{w,\mathfrak{c}}(v_{\mathrm{tot}})_{\mathbb{C}_{p}}\times \mathcal{Z}\right). \end{align*} $$
This concludes the proof.
4.5.2 Gluing around cusps
Consider the unit disc
$\mathcal {D}_{\mathbb {C}_{p}}$
over
$\mathrm {Sp}\left (\mathbb {C}_{p}\right )$
centered at the origin O. Put
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}=\mathcal {D}_{\mathbb {C}_{p}}\setminus \{O\}$
.
Lemma 4.16. Let
$\mathcal {Z}$
be a quasi-compact reduced rigid analytic variety over
$\mathbb {C}_{p}$
. Then the ring
$\mathcal {O}\left (\mathcal {Z}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
can be identified with the ring
$R_{\mathcal {Z}}$
of power series
$\sum _{n\in \mathbb {Z}}a_{n} T^{n}$
with
$a_{n}\in \mathcal {O}(\mathcal {Z})$
such that
$$ \begin{align} \lim_{n\to +\infty} \sup_{z\in \mathcal{Z}} \lvert a_{n}(z)\rvert=0,\qquad \lim_{n\to +\infty} \sup_{z\in \mathcal{Z}} \lvert a_{-n}(z)\rvert\rho^{-n}=0 \end{align} $$
for any rational number
$\rho $
satisfying
$0<\rho \leq 1$
.
Proof. For any nonnegative rational number
$\rho \leq 1$
, let
$\mathcal {A}[\rho ,1]_{\mathbb {C}_{p}}$
be the closed annulus with parameter T over
$\mathbb {C}_{p}$
defined by
$\rho \leq \lvert T\rvert \leq 1$
. Then we have an admissible covering
$$ \begin{align*} \mathcal{D}^{\times}_{\mathbb{C}_{p}}=\bigcup_{\rho\to 0+} \mathcal{A}[\rho,1]_{\mathbb{C}_{p}} \end{align*} $$
of
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}$
.
First suppose that
$\mathcal {Z}=\mathrm {Sp}(A)$
is affinoid. Let
$\rho $
be a rational number satisfying
$0<\rho \leq 1$
. We denote by
$\lvert -\rvert _{\sup }$
the supremum norm on A. Since A is reduced, [Reference Bosch, Güntzer and Remmert10, Theorem 6.2.4/1] shows that
$\lvert -\rvert _{\sup }$
defines the Banach topology on A. Then the ring
$\mathcal {O}\left (\mathrm {Sp}(A)\times \mathcal {A}[\rho ,1]_{\mathbb {C}_{p}}\right )$
can be identified with the ring of power series
$\sum _{n\in \mathbb {Z}}a_{n} T^{n}$
with
$a_{n}\in A$
such that
$$ \begin{align} \lim_{n\to +\infty} \lvert a_{n}\rvert_{\sup}=0,\qquad \lim_{n\to +\infty} \lvert a_{-n}\rvert_{\sup}\rho^{-n}=0. \end{align} $$
This identification is compatible with the restriction to
$\mathcal {A}[\rho ^{\prime },1]_{\mathbb {C}_{p}}$
for any
$\rho <\rho ^{\prime }$
, and also with the restriction to any affinoid subdomain of
$\mathrm {Sp}(A)$
. In particular, the natural map
$$ \begin{align*} \mathcal{O}\left(\mathrm{Sp}(A)\times\mathcal{A}[\rho,1]_{\mathbb{C}_{p}}\right)\to \mathcal{O}\left(\mathrm{Sp}(A)\times\mathcal{A}[\rho^{\prime},1]_{\mathbb{C}_{p}}\right) \end{align*} $$
is injective for any
$\rho <\rho ^{\prime }$
and thus
$$ \begin{align*} \mathcal{O}\left(\mathrm{Sp}(A)\times\mathcal{D}^{\times}_{\mathbb{C}_{p}}\right)=\bigcap_{\rho\to 0+} \mathcal{O}\left(\mathrm{Sp}(A)\times\mathcal{A}[\rho,1]_{\mathbb{C}_{p}}\right). \end{align*} $$
This yields an isomorphism
$\Phi _{\mathcal {Z}}:\mathcal {O}\left (\mathcal {Z}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )\simeq R_{\mathcal {Z}}$
when
$\mathcal {Z}$
is affinoid, which is compatible with the restriction to any affinoid subdomain of
$\mathcal {Z}$
.
For
$\mathcal {Z}$
not necessarily affinoid, take a finite admissible affinoid covering
$\mathcal {Z}=\bigcup _{i\in I} \mathcal {U}_{i}$
with
$\mathcal {U}_{i}=\mathrm {Sp}(A_{i})$
. Take
$f\in \mathcal {O}\left (\mathcal {Z}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
and write
$$ \begin{align*} \Phi_{\mathcal{U}_{i}}\left(f\rvert_{\mathcal{U}_{i}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}}\right)=\sum_{n\in \mathbb{Z}} a_{i,n} T^{n} \end{align*} $$
with
$a_{i,n}\in A_{i}$
. Then
$a_{i,n}$
s can be glued to obtain an element
$a_{n}\in \mathcal {O}(\mathcal {Z})$
. Put
$\Phi (f)=\sum _{n\in \mathbb {Z}} a_{n} T^{n}$
. Since I is a finite set, we can check that
$a_{n}$
s also satisfy equation (4.14) and thus
$\Phi (f)\in R_{\mathcal {Z}}$
. On the other hand, for any element
$g=\sum _{n\in \mathbb {Z}} a_{n} T^{n}$
of
$R_{\mathcal {Z}}$
, put
$\Psi (g)_{i}=\sum _{n\in \mathbb {Z}} a_{n}\rvert _{\mathcal {U}_{i}} T^{n}$
. Then
$\Psi (g)_{i}\in R_{\mathcal {U}_{i}}$
and the elements
$\Phi _{\mathcal {U}_{i}}^{-1}(\Psi (g)_{i})\in \mathcal {O}\left (\mathcal {U}_{i}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
can be glued to obtain an element
$\Psi (g)\in \mathcal {O}\left (\mathcal {Z}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
. Then
$\Phi $
and
$\Psi $
are inverse to each other and the lemma follows.
Next we show the following variant of [Reference Buzzard and Calegari13, Lemma 7.1]:
Lemma 4.17. Let
$\mathcal {Z}$
be a quasi-compact smooth rigid analytic variety over
$\mathbb {C}_{p}$
. Let
$\mathcal {V}$
be an admissible open subset of
$\mathcal {Z}$
which meets every connected component of
$\mathcal {Z}$
. Let f be an element of
$\mathcal {O}\left (\mathcal {Z}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
. Suppose that
$f\rvert _{\mathcal {V}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}}$
extends to an element of
$\mathcal {O}\left (\mathcal {V} \times \mathcal {D}_{\mathbb {C}_{p}}\right )$
. Then f extends to an element of
$\mathcal {O}\left (\mathcal {Z}\times \mathcal {D}_{\mathbb {C}_{p}}\right )$
.
Proof. By taking an admissible affinoid open subset of the intersection of
$\mathcal {V}$
and each connected component of
$\mathcal {Z}$
and replacing
$\mathcal {V}$
with their union, we may assume that
$\mathcal {V}$
is quasi-compact. By Lemma 3.1, the assumption on
$\mathcal {V}$
yields injections
$$ \begin{align*} \mathcal{O}(\mathcal{Z})\to \mathcal{O}(\mathcal{V}),\qquad \mathcal{O}\left(\mathcal{Z}\times\mathcal{D}^{\times}_{\mathbb{C}_{p}}\right) \to \mathcal{O}\left(\mathcal{V}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}\right) \gets \mathcal{O}\left(\mathcal{V}\times \mathcal{D}_{\mathbb{C}_{p}}\right). \end{align*} $$
From Lemma 4.16, we see that the intersection of
$\mathcal {O}\left (\mathcal {Z}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
and
$\mathcal {O}\left (\mathcal {V}\times \mathcal {D}_{\mathbb {C}_{p}}\right )$
inside
$\mathcal {O}\left (\mathcal {V}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
is the set of formal power series
$\sum _{n\geq 0} a_{n} T^{n}$
with
$a_{n}\in \mathcal {O}(\mathcal {Z})$
satisfying
$$ \begin{align*} \lim_{n\to +\infty} \sup_{z\in \mathcal{Z}} \lvert a_{n}(z)\rvert=0, \end{align*} $$
which is equal to
$\mathcal {O}\left (\mathcal {Z}\times \mathcal {D}_{\mathbb {C}_{p}}\right )$
.
Lemma 4.18.
$$ \begin{align*} \mathcal{O}^{\circ}\left(\mathcal{D}^{\times}_{\mathbb{C}_{p}}\right)\subseteq \mathcal{O}\left(\mathcal{D}_{\mathbb{C}_{p}}\right). \end{align*} $$
Proof. Let
$f=\sum _{n\in \mathbb {Z}} a_{n} T^{n}$
be an element of
$\mathcal {O}^{\circ }\left (\mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
. Consider the Newton polygon of f. Then the assumption implies that any point
$\left (n,v_{p}(a_{n})\right )$
lies above the line
$y=-rx$
for any nonnegative rational number r, which forces
$a_{n}=0$
for any
$n<0$
.
Proposition 4.19. Let
$\varphi :\mathcal {D}^{\times }_{\mathbb {C}_{p}}\to (\mathcal {E}\rvert _{\mathcal {X}})_{\mathbb {C}_{p}}$
be a morphism of rigid analytic varieties over
$\mathbb {C}_{p}$
such that the composite
$\mathcal {D}^{\times }_{\mathbb {C}_{p}} \to (\mathcal {E}\rvert {}_{\mathcal {X}})_{\mathbb {C}_{p}}\to \mathcal {X}_{\mathbb {C}_{p}}$
extends to an n-analytic map
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {X}_{\mathbb {C}_{p}}$
. Let
$\left (\nu ^{\mathcal {D}_{\mathbb {C}_{p}}},w^{\mathcal {D}_{\mathbb {C}_{p}}}\right )$
be the weight associated to the map
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {X}_{\mathbb {C}_{p}}$
. Suppose that for some nonnegative rational number
$v^{\prime }<(p-1)/p^{n}$
, we are given an element
$$ \begin{align*} f=(f_{\mathfrak{c}})_{\mathfrak{c}\in\left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}}\in \bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}\right) \end{align*} $$
and an admissible affinoid covering
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}=\bigcup _{i\in I} \mathcal {U}_{i}$
such that the restriction
$f\rvert _{\mathcal {U}_{i}}$
for each
$i\in I$
is an eigenform of
$S^{G}\left (\mu _{N}, \left (\nu ^{\mathcal {U}_{i}},w^{\mathcal {U}_{i}}\right )\right )(v^{\prime })$
with eigensystem
$\varphi ^{*}: \mathbf {T}\to \mathcal {O}\left (\mathcal {D}_{\mathbb {C}_{p}}^{\times }\right )\to \mathcal {O}(\mathcal {U}_{i})$
and
$f(z)$
is normalized for any
$z\in \mathcal {D}^{\times }_{\mathbb {C}_{p}}$
. Then there exists an eigenform
$f^{\prime }\in S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {D}_{\mathbb {C}_{p}}},w^{\mathcal {D}_{\mathbb {C}_{p}}}\right )\right )(v^{\prime })$
such that
$f^{\prime }(z)$
is normalized for any
$z\in \mathcal {D}_{\mathbb {C}_{p}}$
, and it is an eigenform with eigensystem
$\varphi ^{*}(z): \mathbf {T}\to \mathcal {O}\left (\mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )\to \mathbb {C}_{p}$
for any
$z\in \mathcal {D}^{\times }_{\mathbb {C}_{p}}$
.
Proof. Consider the map
$\pi _{w}^{\mathrm {rig}}: \mathcal {IW}_{w,\mathfrak {c}}^{+}(v^{\prime })_{\mathbb {C}_{p}} \to \bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v^{\prime })_{\mathbb {C}_{p}}$
as before. Let
$\mathcal {S}_{\mathfrak {c}}$
be an admissible affinoid open subset of
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v^{\prime })_{\mathbb {C}_{p}}$
as in Corollary 4.12(1). Put
$\mathcal {I}_{\mathfrak {c}}=\left (\pi _{w}^{\mathrm {rig}}\right )^{-1}(\mathcal {S}_{\mathfrak {c}})$
. Then
$\mathcal {I}_{\mathfrak {c}}$
is an admissible open subset which meets every connected component of
$\mathcal {IW}_{w,\mathfrak {c}}^{+}(v^{\prime })_{\mathbb {C}_{p}}$
such that
$f_{\mathfrak {c}}(z)\rvert _{\mathcal {I}_{\mathfrak {c}}}$
has absolute value bounded by
$1$
for any
$z\in \mathcal {D}^{\times }_{\mathbb {C}_{p}}$
. Hence
$f_{\mathfrak {c}}\rvert _{\mathcal {I}_{\mathfrak {c}}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}}$
also has absolute value bounded by
$1$
.
Note that
$\mathcal {I}_{\mathfrak {c}}$
is quasi-compact, since
$\pi _{w}$
is quasi-compact. By Lemma 4.16, we can write
$$ \begin{align*} f_{\mathfrak{c}}\rvert_{\mathcal{I}_{\mathfrak{c}}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}}=\sum_{n\in \mathbb{Z}} a_{n} T^{n} \end{align*} $$
with some
$a_{n}\in \mathcal {O}(\mathcal {I}_{\mathfrak {c}})$
. Lemma 4.18 implies
$a_{n}(x)=0$
for any
$x\in \mathcal {I}_{\mathfrak {c}}$
and any
$n<0$
. Since
$\mathcal {I}_{\mathfrak {c}}$
is reduced, we obtain
$a_{n}=0$
for any
$n<0$
and thus
$$ \begin{align*} f_{\mathfrak{c}}\rvert_{\mathcal{I}_{\mathfrak{c}}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}}\in \mathcal{O}\left(\mathcal{I}_{\mathfrak{c}}\times \mathcal{D}_{\mathbb{C}_{p}}\right). \end{align*} $$
Therefore, by Lemma 4.17 we see that
$f_{\mathfrak {c}}$
extends to an element
$\tilde {f}_{\mathfrak {c}}$
of
$\mathcal {O}\left (\mathcal {IW}_{w,\mathfrak {c}}^{+}(v^{\prime })_{\mathbb {C}_{p}}\times\mathcal {D}_{\mathbb {C}_{p}}\right )$
.
Write
$\mathcal {D}_{\mathbb {C}_{p}}=\mathrm {Sp}\left (\mathbb {C}_{p}\langle T\rangle \right )$
. Note that the ring
$\mathcal {O}\left (\mathcal {IW}_{w,\mathfrak {c}}^{+}(v^{\prime })_{\mathbb {C}_{p}}\times \mathcal {D}_{\mathbb {C}_{p}}\right )$
is T-torsion free. We claim that if
$f_{\mathfrak {c}}\neq 0$
, then there exists a nonnegative integer
$m_{\mathfrak {c}}$
satisfying
$$ \begin{align*} \tilde{f}_{\mathfrak{c}}\in T^{m_{\mathfrak{c}}} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}\times \mathcal{D}_{\mathbb{C}_{p}}\right)\setminus T^{m_{\mathfrak{c}}+1} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}\times \mathcal{D}_{\mathbb{C}_{p}}\right). \end{align*} $$
Indeed, since
$\mathcal {IW}_{w,\mathfrak {c}}^{+}(v^{\prime })_{\mathbb {C}_{p}}$
is smooth, we can take an admissible affinoid covering
$$ \begin{align*} \mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}=\bigcup_{j\in J}\mathcal{V}_{j},\quad \mathcal{V}_{j}=\mathrm{Sp}\left(A_{j}\right), \end{align*} $$
such that every
$A_{j}$
is a Noetherian domain. Suppose that
$$ \begin{align*} \tilde{f}_{\mathfrak{c}}\in \bigcap_{m\geq 0} T^{m} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}\times \mathcal{D}_{\mathbb{C}_{p}}\right). \end{align*} $$
Since
$A_{j}\langle T\rangle $
is also a Noetherian domain, Krull’s intersection theorem implies
$\tilde {f}_{\mathfrak {c}}\rvert _{\mathcal {V}_{j}\times \mathcal {D}_{\mathbb {C}_{p}}}=0$
for any
$j\in J$
and thus
$\tilde {f}_{\mathfrak {c}}=0$
, which is a contradiction.
Put
$m=\min \left \{m_{\mathfrak {c}}\mid \mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)},f_{\mathfrak {c}}\neq 0\right \}$
. Let
$\tilde {f}^{\prime }_{\mathfrak {c}}$
be the unique element of
$\mathcal {O}\left (\mathcal {IW}_{w,\mathfrak {c}}^{+}(v^{\prime })_{\mathbb {C}_{p}}\times \mathcal {D}_{\mathbb {C}_{p}}\right )$
satisfying
$\tilde {f}_{\mathfrak {c}}=T^{m} \tilde {f}^{\prime }_{\mathfrak {c}}$
. Since the maps
$$ \begin{align*} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}\times \mathcal{D}_{\mathbb{C}_{p}}\right)&\to \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}\right)\\ &\to \prod_{i\in I} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v^{\prime})_{\mathbb{C}_{p}}\times \mathcal{U}_{i}\right) \end{align*} $$
are injective by Lemma 3.1, the element
$\tilde {f}^{\prime }_{\mathfrak {c}}$
is also
$\kappa ^{\mathcal {D}_{\mathbb {C}_{p}}}$
-equivariant and
$\Delta $
-stable. Moreover, note that the restriction map
$\mathcal {O}\left (\mathcal {V}\times \mathcal {D}_{\mathbb {C}_{p}}\right )\to \mathcal {O}\left (\mathcal {V}\times \mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
is injective for any
$\mathbb {C}_{p}$
-affinoid variety
$\mathcal {V}$
. For the boundary divisor D of
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v^{\prime })_{\mathbb {C}_{p}}$
, we have the commutative diagram
where the vertical arrows are injective and the bottom arrow is bijective. This implies that the element
$\tilde {f}^{\prime }_{\mathfrak {c}}$
is a cusp form. Hence the collection
$\tilde {f}^{\prime }=\left (\tilde {f}^{\prime }_{\mathfrak {c}}\right )_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
is an element of
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {D}_{\mathbb {C}_{p}}},w^{\mathcal {D}_{\mathbb {C}_{p}}}\right )\right )(v^{\prime })$
such that
$\tilde {f}^{\prime }(z)\neq 0$
for any
$z\in \mathcal {D}_{\mathbb {C}_{p}}$
.
Let
$\Lambda (\mathfrak {n})$
be the image of
$T_{\mathfrak {n}}$
(resp.,
$S_{\mathfrak {n}}$
) by the map
$\varphi ^{*}: \mathbf {T}\to \mathcal {O}\left (\mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
. By Corollary 4.10, the specialization
$\Lambda (\mathfrak {n})(z)$
is p-integral for any
$z\in \mathcal {D}^{\times }_{\mathbb {C}_{p}}$
. Thus Lemma 4.18 shows
$\Lambda (\mathfrak {n})\in \mathcal {O}\left (\mathcal {D}_{\mathbb {C}_{p}}\right )$
. By the injectivity mentioned above, we see that
$\tilde {f}^{\prime }$
is an eigenform on which
$T_{\mathfrak {n}}$
(resp.,
$S_{\mathfrak {n}}$
) acts by
$\Lambda (\mathfrak {n})$
. Now Lemma 4.13 concludes the proof of the proposition.
5 Properness at integral weights
Let
$\mathcal {E}\to \mathcal {W}^{G}$
be the Hilbert eigenvariety as in §3.3.3. Let
$\mathcal {D}_{\mathbb {C}_{p}}$
be the unit disc over
$\mathrm {Sp}\left (\mathbb {C}_{p}\right )$
centered at the origin O and put
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}=\mathcal {D}_{\mathbb {C}_{p}}\setminus \{O\}$
. In this section, we prove the following main theorem of this paper:
Theorem 5.1. Suppose that F is unramified over p and that for any prime ideal
$\mathfrak {p}\mid p$
of F, the residue degree
$f_{\mathfrak {p}}$
satisfies
$f_{\mathfrak {p}}\leq 2$
(resp., p splits completely in F) for
$p\geq 3$
(resp.,
$p=2$
). Consider a commutative diagram
of rigid analytic varieties over
$\mathbb {C}_{p}$
, where the left vertical arrow is the natural inclusion. Suppose that
$\psi (O)$
is
$1$
-integral (resp.,
$1$
-doubly even) in the sense of §3.3.2. Then there exists a morphism
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {E}_{\mathbb {C}_{p}}$
of rigid analytic varieties over
$\mathbb {C}_{p}$
such that the diagram with this morphism added is also commutative.
Before proving the theorem, we summarize the structure of the proof.
Step 1. By shrinking the disc, we reduce the theorem to the case where
$\psi $
is
$1$
-analytic.Step 2. Using Proposition 4.15, we convert the map
$\varphi $
to an analytic function f on with some small
$$ \begin{align*} \coprod_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathcal{IW}_{w,\mathfrak{c}}^{+}(v_{\mathrm{tot}})_{\mathbb{C}_{p}}\times \mathcal{D}_{\mathbb{C}_{p}}^{\times} \end{align*} $$
$v>0$
such that locally on
$\mathcal {D}_{\mathbb {C}_{p}}^{\times }$
it gives a normalized eigenform with eigensystem determined by
$\varphi $
.
Step 3. By analytic continuation via the
$U_{p}$
-operator, we extend f to By Proposition 4.19, we may assume that f can be extended to the puncture so that it defines a normalized eigenform over
$$ \begin{align*} \coprod_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathcal{IW}_{w,\mathfrak{c}}^{+}\left(\tfrac{1}{p+1}\right)_{\mathbb{C}_{p}}\times \mathcal{D}_{\mathbb{C}_{p}}^{\times}. \end{align*} $$
$\mathcal {D}_{\mathbb {C}_{p}}$
.
Step 4. Using results from §3.2, we show that
$U_{p} f$
can be evaluated on the connected neighborhood
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}$
for any critical point Q, modulo passing to a finite covering
$r: \mathcal {Y}_{\mathfrak {c},p}^{1}\to \mathcal {Y}_{\mathfrak {c},p}$
.Step 5. Supposing
$f(O)$
is of infinite slope, we deduce a contradiction by a combinatorial argument using critical points in the spirit of Buzzard and Calegari. Then Proposition 2.7 yields a desired extension of
$\varphi $
to
$\mathcal {D}_{\mathbb {C}_{p}}$
.
Proof.
Step 1. Let
$e_{1},\dotsc ,e_{g}$
be a basis of the
$\mathbb {Z}_{p}$
-module
$2p\left (\mathcal {O}_{F}\otimes \mathbb {Z}_{p}\right )$
and put
$E_{i}=\exp (e_{i})\in 1+2p\left (\mathcal {O}_{F}\otimes \mathbb {Z}_{p}\right )$
. Similarly, let
$e_{g+1}$
be a basis of the
$\mathbb {Z}_{p}$
-module
$2p\mathbb {Z}_{p}$
and put
$E_{g+1}=\exp \left (e_{g+1}\right )\in 1+2p\mathbb {Z}_{p}$
. Let
$(\nu ^{\mathrm {un}}, w^{\mathrm {un}})$
be the universal character on
$\mathcal {W}^{G}$
. Note that
$\mathcal {W}^{G}_{\mathbb {C}_{p}}$
is the disjoint union of finitely many copies of the open unit polydisc defined by
$$ \begin{align*} \lvert X_{1}\rvert<1,\dotsc, \left\lvert X_{g+1}\right\rvert<1 \end{align*} $$
with parameters
$X_{1},\dotsc ,X_{g+1}$
: the connected components are parametrized by the finite-order characters
$$ \begin{align*} \varepsilon:\mathbb{T}(\mathbb{Z}/2 p\mathbb{Z})\times (\mathbb{Z}/2 p\mathbb{Z})^{\times} \to \mathcal{O}_{\mathbb{C}_{p}}^{\times} \end{align*} $$
and on each connected component, the point defined by
$X_{i}\mapsto x_{i}$
corresponds to the character
$(\nu ,w)$
satisfying
$\nu (E_{i})=1+x_{i}$
for any
$i\leq g$
, and
$w\left (E_{g+1}\right )=1+x_{g+1}$
.
Put
$q=p$
if
$p\geq 3$
and
$q=16$
if
$p=2$
. Since
$\psi (O)$
is
$1$
-integral, it comes from a K-valued point of
$\mathcal {W}^{G}$
, which we also denote by
$\psi (O)$
. This corresponds to a finite-order character
$\varepsilon _{O}$
and a map
$X_{i}\mapsto x_{i}$
with some
$x_{i}\in q\mathcal {O}_{K}$
. For
$p=2$
, the assumption that
$\psi (O)$
is
$1$
-doubly even implies that
$\varepsilon _{O}$
is trivial on the torsion subgroup of
$1+2(\mathcal {O}_{F}\otimes \mathbb {Z}_{2})$
. Put
$E^{\prime }_{i}=(-1)^{p-1}E_{i}$
. The group
$1+p\left (\mathcal {O}_{F}\otimes \mathbb {Z}_{p}\right )$
is topologically generated by
$E_{i}$
s and
$E^{\prime }_{i}$
s. We have
$$ \begin{align*} (\nu^{\mathrm{un}}, w^{\mathrm{un}})(E_{i})=(\nu^{\mathrm{un}}, w^{\mathrm{un}})\left(E^{\prime}_{i}\right)=1+X_{i} \end{align*} $$
on the
$\varepsilon _{O}$
-component of
$\mathcal {W}^{G}$
. Let
$\mathcal {U}=\mathrm {Sp}(R)$
be the admissible affinoid open subset of the
$\varepsilon _{O}$
-component of
$\mathcal {W}^{G}$
defined by
$\lvert X_{i}-x_{i}\rvert \leq \lvert q\rvert $
for any i. Then
$1+X_{i}=1+x_{i}+(X_{i}-x_{i})\in 1+qR^{\circ }$
and the universal character
$(\nu ^{\mathrm {un}}, w^{\mathrm {un}})$
is
$1$
-analytic on
$\mathcal {U}$
.
We denote by
$\mathcal {D}_{\rho ,\mathbb {C}_{p}}$
the closed disc of radius
$\rho $
centered at the origin over
$\mathbb {C}_{p}$
. Consider the element
$\psi ^{*}(X_{i})(T)$
of the ring
$\mathcal {O}\left (\mathcal {D}_{\mathbb {C}_{p}}\right )=\mathbb {C}_{p}\langle T\rangle $
. Since
$\psi ^{*}(X_{i})(0)=x_{i}$
, there exists a positive rational number
$\rho <1$
such that
$$ \begin{align*} \lvert t\rvert\leq \rho \Rightarrow \lvert\psi^{*}(X_{i})(t)-x_{i}\rvert\leq \lvert q\rvert \end{align*} $$
for any i. This means
$\psi \left (\mathcal {D}_{\rho ,\mathbb {C}_{p}}\right )\subseteq \mathcal {U}_{\mathbb {C}_{p}}$
. If we can construct a morphism
$\mathcal {D}_{\rho ,\mathbb {C}_{p}}\to \mathcal {E}_{\mathbb {C}_{p}}$
which makes the diagram in the theorem commutative, then by gluing we obtain the desired map
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {E}_{\mathbb {C}_{p}}$
. Thus, by shrinking the disc, we may assume that
$\psi $
factors through
$\mathcal {U}_{\mathbb {C}_{p}}$
.
Step 2. Put
$n=1$
and
$v=v_{1}$
. We may assume
$v<1/(p+1)$
so that we have
$$ \begin{align*} \bar{\mathcal{M}}(\mu_{N},\mathfrak{c})(v_{\mathrm{tot}})\subseteq\bar{\mathcal{M}}(\mu_{N},\mathfrak{c})\left(\tfrac{1}{p+1}\right). \end{align*} $$
By Remark 2.4, the rigid analytic variety
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}$
is principally refined. Applying Proposition 4.15 to the map
$\varphi : \mathcal {D}^{\times }_{\mathbb {C}_{p}}\to (\mathcal {E}\rvert _{\mathcal {U}})_{\mathbb {C}_{p}}$
, we obtain an element
$$ \begin{align*} f\in \bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathcal{O}\left(\mathcal{IW}_{w,\mathfrak{c}}^{+}(v_{\mathrm{tot}})_{\mathbb{C}_{p}}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}\right) \end{align*} $$
and an admissible affinoid covering
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}=\bigcup _{i\in I} \mathcal {U}_{i}$
such that the restriction
$f\rvert _{\mathcal {U}_{i}}$
for each
$i\in I$
is an eigenform of
$S^{G}\left (\mu _{N}, \left (\nu ^{\mathcal {U}_{i}},w^{\mathcal {U}_{i}}\right )\right )(v_{\mathrm {tot}})$
with eigensystem
$\varphi ^{*}: \mathbf {T}\to \mathcal {O}\left (\mathcal {D}_{\mathbb {C}_{p}}^{\times }\right )\to \mathcal {O}(\mathcal {U}_{i})$
and
$f(z)$
is normalized for any
$z\in \mathcal {D}^{\times }_{\mathbb {C}_{p}}$
.
Step 3. Since
$\varphi ^{*}$
comes from the eigenvariety
$\mathcal {E}$
, the
$U_{p}$
-eigenvalue
$\varphi ^{*}\left (U_{p}\right )\in \mathcal {O}(\mathcal {U}_{i})$
of
$f\rvert _{\mathcal {U}_{i}}$
satisfies
$\varphi ^{*}\left (U_{p}\right )(z)\neq 0$
for any
$z\in \mathcal {U}_{i}\left (\mathbb {C}_{p}\right )$
, and thus we have
$\varphi ^{*}\left (U_{p}\right )\in \mathcal {O}(\mathcal {U}_{i})^{\times }$
. Since
$U_{p}$
improves the overconvergence from v to
$pv$
, taking
$\varphi ^{*}\left (U_{p}\right )^{-1}U_{p}\left (f\rvert _{\mathcal {U}_{i}}\right )$
repeatedly we can find an eigenform
$$ \begin{align*} g_{i} \in S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}_{i}},w^{\mathcal{U}_{i}}\right)\right)\left(\tfrac{1}{p+1}\right) \end{align*} $$
with eigensystem
$\varphi ^{*}: \mathbf {T}\to \mathcal {O}\left (\mathcal {D}_{\mathbb {C}_{p}}^{\times }\right )\to \mathcal {O}(\mathcal {U}_{i})$
which extends
$f\rvert _{\mathcal {U}_{i}}$
. Note that for any
$z\in \mathcal {U}_{i}\left (\mathbb {C}_{p}\right )$
we have a commutative diagram
where the horizontal arrows are the restriction maps and the vertical arrows are the specialization maps. This implies that the specialization
$g_{i}(z)$
is also nonzero for any
$z\in \mathcal {U}_{i}\left (\mathbb {C}_{p}\right )$
. Since the q-expansion is determined by the restriction to the ordinary locus,
$g_{i}(z)$
is also normalized for any
$z\in \mathcal {U}_{i}\left (\mathbb {C}_{p}\right )$
. Since the Hecke eigenvalues of
$g_{i}(z)$
are also given by the eigensystem
$\varphi ^{*}(z): \mathbf {T}\to \mathcal {O}(\mathcal {U}_{i})\overset {z^{*}}{\to } \mathbb {C}_{p}$
, a gluing argument as in the proof of Proposition 4.15 shows that
$g_{i}$
s can be glued. In other words, we may assume
$$ \begin{align*} f\in \bigoplus_{\mathfrak{c}\in \left[\mathrm{Cl}^{+}(F)\right]^{\left(p\right)}} \mathcal{O}\left(\mathcal{IW}^{+}_{w,\mathfrak{c}}\left(\tfrac{1}{p+1}\right)_{\mathbb{C}_{p}}\times \mathcal{D}^{\times}_{\mathbb{C}_{p}}\right). \end{align*} $$
By Proposition 4.19, we may replace f by an eigenform of the space
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {D}_{\mathbb {C}_{p}}},w^{\mathcal {D}_{\mathbb {C}_{p}}}\right )\right )\left (\tfrac {1}{p+1}\right )$
such that every specialization on
$\mathcal {D}_{\mathbb {C}_{p}}$
is normalized, which we also denote by
$f=(f_{\mathfrak {c}})_{\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{\left (p\right )}}$
. By Lemma 3.13, we have an isomorphism
$$ \begin{align*} S^{G}\left(\mu_{N},\left(\nu^{\mathcal{U}},w^{\mathcal{U}}\right)\right)(v_{\mathrm{tot}})\hat{\otimes}_{R,z^{*}} k(z) \simeq S^{G}\left(\mu_{N},\left(\nu^{\mathcal{D}_{\mathbb{C}_{p}}}(z),w^{\mathcal{D}_{\mathbb{C}_{p}}}(z)\right)\right)(v_{\mathrm{tot}}) \end{align*} $$
for any
$z\in \mathcal {D}_{\mathbb {C}_{p}}$
. Thus the map
$\mathbf {T}\to \mathcal {O}\left (\mathcal {D}_{\mathbb {C}_{p}}\right )$
defined by the eigenvalues of f is a family of eigensystems in
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v_{\mathrm {tot}})$
over
$\mathcal {D}_{\mathbb {C}_{p}}$
such that its restriction to
$\mathcal {D}^{\times }_{\mathbb {C}_{p}}$
is
$\varphi ^{*}:\mathbf {T}\to \mathcal {O}\left (\mathcal {D}^{\times }_{\mathbb {C}_{p}}\right )$
. In particular, it is of finite slopes over
$\mathcal {D}_{\mathbb {C}_{p}}^{\times }$
. If
$f(O)$
is of finite slope, then Proposition 2.7 yields a morphism
$\mathcal {D}_{\mathbb {C}_{p}}\to \mathcal {E}\rvert _{\mathcal {U}_{\mathbb {C}_{p}}}$
with the desired property.
Step 4. Put
$\psi (O)=(\nu (O),w(O))$
and
$\kappa =k(\nu (O),w(O))$
, which are
$1$
-integral by assumption. Let
$ \kappa _{1}=\left (k_{\beta }\right )_{\beta \in \mathbb {B}_{F}}$
be the integral weight corresponding to the restriction
$\kappa \rvert _{\mathbb {T}^{0}_{1}\left (\mathbb {Z}_{p}\right )}$
. For any nonzero fractional ideal
$ \mathfrak {c}$
of F, let
$\mathcal {X}_{\mathfrak {c}}=\mathfrak {M}(\mu _{N},\mathfrak {c})^{\mathrm {rig}}$
be the Raynaud generic fiber of the p-adic formal completion
$\mathfrak {M}(\mu _{N},\mathfrak {c})$
of
$M(\mu _{N},\mathfrak {c})$
. We also write
$\mathcal {X}_{\mathfrak {c}}(v^{\prime })=\mathfrak {M}(\mu _{N},\mathfrak {c})(v^{\prime })^{\mathrm {rig}}$
for any
$v^{\prime }<1$
. For any
$v^{\prime }<(p-1)/p$
and the morphism
$h_{1}:\bar {\mathfrak {M}}(\Gamma _{1}(p),\mu _{N},\mathfrak {c})(v^{\prime })\to \bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v^{\prime })$
, we put
$h=h_{1}^{\mathrm {rig}}$
and
$\mathcal {X}_{\mathfrak {c}}^{1}(v^{\prime })=h^{-1}(\mathcal {X}_{\mathfrak {c}}(v^{\prime }))$
.
Consider the rigid analytic variety
$\mathcal {Y}_{\mathfrak {c},p}$
as in §3.2 and the natural projection
$\pi :\mathcal {Y}_{\mathfrak {c},p}\to \mathcal {X}_{\mathfrak {c}}$
. Put
$\mathcal {Y}_{\mathfrak {c},p}(v^{\prime })=\pi ^{-1}(\mathcal {X}_{\mathfrak {c}}(v^{\prime }))$
. For the universal p-cyclic subgroup scheme
$H^{\mathrm {un}}$
over
$\mathcal {Y}_{\mathfrak {c},p}$
, we put
$$ \begin{align*} \mathcal{Y}^{1}_{\mathfrak{c},p}=\mathrm{Isom}_{\mathcal{Y}_{\mathfrak{c},p}}\left(\mathcal{D}_{F}^{-1}\otimes \mu_{p}, H^{\mathrm{un}}\right). \end{align*} $$
We denote by r the natural projection
$\mathcal {Y}^{1}_{\mathfrak {c},p}\to \mathcal {Y}_{\mathfrak {c},p}$
. Put
$\pi ^{1}=\pi \circ r$
and
$\mathcal {Y}^{1}_{\mathfrak {c},p}(v^{\prime })=\left (\pi ^{1}\right )^{-1}(\mathcal {X}_{\mathfrak {c}}(v^{\prime }))$
. We write the base extensions to
$\mathbb {C}_{p}$
of these maps also as h,
$\pi $
, r, and
$\pi ^{1}$
, respectively. We consider
$\mathcal {U}^{1}_{\mathfrak {c}}:=\mathcal {X}_{\mathfrak {c}}^{1}\left (\tfrac {1}{p+1}\right )$
as a Zariski open subset of
$\mathcal {Y}^{1}_{\mathfrak {c},p}\left (\tfrac {1}{p+1}\right )$
. Then we have an isomorphism
$h^{*}\Omega ^{\kappa }\simeq \left (\pi ^{1}\right )^{*}\Omega ^{\kappa }\rvert _{\mathcal {U}^{1}_{\mathfrak {c},\mathbb {C}_{p}}}$
. Note that the sheaf
$h^{*}\Omega ^{\kappa }$
in this case of
$1$
-integral weight is isomorphic to the sheaf
$h^{*}\Omega ^{\kappa _{1}}\simeq h^{*}\omega ^{\kappa _{1}} {}_{\bar {\mathcal {A}}^{\mathrm {un}},\mathbb {C}_{p}}$
as in §3.3.1. The sheaf
$\left (\pi ^{1}\right )^{*}\Omega ^{\kappa _{1}}$
is defined over the whole rigid analytic variety
$\mathcal {Y}^{1}_{\mathfrak {c},p,\mathbb {C}_{p}}$
and satisfies
$\left (\pi ^{1}\right )^{*}\Omega ^{\kappa _{1}}\rvert _{\mathcal {U}^{1}_{\mathfrak {c},\mathbb {C}_{p}}}\simeq h^{*}\Omega ^{\kappa }$
. Thus, when
$\mathfrak {c}$
lies in
$\left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
, the element
$f_{\mathfrak {c}}(O)$
defines
$$ \begin{align*} g_{\mathfrak{c}}:=h^{*}f_{\mathfrak{c}}(O)=\left(\pi^{1}\right)^{*}f_{\mathfrak{c}}(O)\rvert_{\mathcal{U}^{1}_{\mathfrak{c},\mathbb{C}_{p}}}\in H^{0}\left(\mathcal{U}^{1}_{\mathfrak{c},\mathbb{C}_{p}}, \left(\pi^{1}\right)^{*}\Omega^{\kappa_{1}}(-D)\right), \end{align*} $$
on which any element a of the Galois group
$\mathbb {T}(\mathbb {Z}/p\mathbb {Z})$
of
$h:\mathcal {U}^{1}_{\mathfrak {c}}\to \mathcal {X}_{\mathfrak {c}}\left (\tfrac {1}{p+1}\right )$
acts trivially. Namely, for any point
$[(A,u)]$
of
$\mathcal {U}^{1}_{\mathfrak {c}}\left (\bar {\mathbb {Q}}_{p}\right )$
with an
$\mathcal {O}_{F}$
-closed immersion
$u:\mathcal {D}_{F}^{-1}\otimes \mu _{p}\to A$
and any
$a\in \mathbb {T}(\mathbb {Z}/p\mathbb {Z})$
, we have
$g_{\mathfrak {c}}(A,au)=g_{\mathfrak {c}}(A,u)$
.
Moreover, we define
$Z^{1}_{\mathfrak {c},p}$
as the scheme over K classifying triples
$(A,u,D)$
consisting of an HBAV A over a base scheme over K with
$\mathfrak {c}$
-polarization, an
$\mathcal {O}_{F}$
-closed immersion
$u: \mathcal {D}_{F}^{-1}\otimes \mu _{p}\to A$
, and a finite flat closed
$\mathcal {O}_{F}$
-subgroup scheme D which is étale locally isomorphic to
$\underline {\mathcal {O}_{F}/p\mathcal {O}_{F}}$
satisfying
$\mathrm {Im}(u)\cap D=0$
. We denote by
$\mathcal {Z}^{1}_{\mathfrak {c},p}$
the analytification of
$Z^{1}_{\mathfrak {c},p}$
restricted to
$\mathcal {X}_{\mathfrak {c}}$
. We have two projections
$$ \begin{align*} q_{1}:\mathcal{Z}^{1}_{p^{-1}\mathfrak{c},p}&\to \mathcal{Y}^{1}_{p^{-1}\mathfrak{c},p} & q_{2}:\mathcal{Z}^{1}_{p^{-1}\mathfrak{c},p}&\to \mathcal{Y}^{1}_{\mathfrak{c},p}\\ (A,u,D)&\mapsto (A,u), & (A,u,D)&\mapsto \left(A/D,\bar{u}\right), \end{align*} $$
where
$\bar {u}$
is the image of u in
$A/D$
. Put
$\mathcal {Z}^{1}_{\mathfrak {c},p}(v^{\prime })=q_{1}^{-1}\left (\mathcal {Y}^{1}_{\mathfrak {c},p}(v^{\prime })\right )$
.
We denote the restriction of the rigid analytic variety
$\mathcal {Y}^{\prime }_{\mathfrak {c},p}\left (\tfrac {1}{p+1}\right )$
defined in §3.3.3 to
$\mathcal {X}_{\mathfrak {c}}\left (\tfrac {1}{p+1}\right )$
also by
$\mathcal {Y}^{\prime }_{\mathfrak {c},p}\left (\tfrac {1}{p+1}\right )$
. We have a finite étale morphism
$$ \begin{align*} \Pi: q_{1}^{-1}\left(\mathcal{U}^{1}_{p^{-1}\mathfrak{c}}\right)\to \mathcal{Y}^{\prime}_{p^{-1}\mathfrak{c},p}\left(\tfrac{1}{p+1}\right),\quad (A,u,H)\mapsto (A,H). \end{align*} $$
The base extensions of these maps to
$\mathbb {C}_{p}$
are also denoted by
$q_{1}$
,
$q_{2}$
, and
$\Pi $
, respectively. By [Reference Hattori29, Corollary 5.3(1)], we have
$q_{1}^{-1}\left (\mathcal {U}^{1}_{p^{-1}\mathfrak {c}}\right )\subseteq q_{2}^{-1}\left (\mathcal {U}^{1}_{\mathfrak {c}}\right )$
and thus
$q_{1}^{-1}\left (\mathcal {U}^{1}_{p^{-1}\mathfrak {c},\mathbb {C}_{p}}\right ) \subseteq q_{2}^{-1}\left (\mathcal {U}^{1}_{\mathfrak {c},\mathbb {C}_{p}}\right )$
. This yields commutative diagrams
where the latter is cartesian.
We have an isomorphism of changing polarizations
$$ \begin{align*} l_{p}: M(\mu_{N},\mathfrak{c})\to M\left(\mu_{N},p^{-1}\mathfrak{c}\right),\quad (A,\lambda)\mapsto (A,p\lambda), \end{align*} $$
which induces an isomorphism
$\mathcal {X}_{\mathfrak {c}}\left (\tfrac {1}{p+1}\right )\to \mathcal {X}_{p^{-1}\mathfrak {c}}\left (\tfrac {1}{p+1}\right )$
and similarly for other rigid analytic varieties defined already. Let
$x_{\mathfrak {p}}\in F^{\times ,+}$
be the element we fixed in §3.3.3 to define
$U_{p}$
and put
$x=p^{-1}\prod _{\mathfrak {p}\mid p}x_{\mathfrak {p}}\in F^{\times ,+,(p)}$
. Then
$l_{p}$
also induces an isomorphism
$$ \begin{align*} \nu(O)(x)l_{p}^{*}: H^{0}\left(\mathcal{X}_{p^{-1}\mathfrak{c}}\left(\tfrac{1}{p+1}\right)_{\mathbb{C}_{p}},\Omega^{\kappa}\right)\to H^{0}\left(\mathcal{X}_{\mathfrak{c}}\left(\tfrac{1}{p+1}\right)_{\mathbb{C}_{p}},\Omega^{\kappa}\right), \end{align*} $$
which is compatible with the map
$L_{x}\circ \prod _{\mathfrak {p}\mid p}x_{\mathfrak {p}}^{*}$
on
$S^{G}\left (\mu _{N},p^{-1}\mathfrak {c},(\nu (O),w(O))\right )\left (\tfrac {1}{p+1}\right )$
. Note that under the identification (4.1), by composing
$L_{x}$
we identify the operator
$U_{p}$
with a tuple of endomorphisms on
$S^{G}(\mu _{N},\mathfrak {c},(\nu (O),w(O)))\left (\tfrac {1}{p+1}\right )$
for each
$\mathfrak {c}\in \left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
.
Take any point
$Q=[(A,\mathcal {H})]\in Y_{\mathfrak {c},p}(\mathcal {O}_{L})$
with some finite extension
$L/K$
such that
$\mathrm {Hdg}_{\beta }(A)=p/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
, which exists by Lemma 3.6. Consider the admissible open subsets
$\mathcal {V}_{\mathfrak {c},Q}$
,
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}$
, and
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}\left (\tfrac {1}{p+1}\right )$
of
$\mathcal {Y}_{\mathfrak {c},p, \mathbb {C}_{p}}$
defined in §3.2. For the point
$l_{p}(Q)\in Y_{p^{-1}\mathfrak {c},p}(\mathcal {O}_{L})$
, we also have similar admissible open subsets
$\mathcal {V}_{p^{-1}\mathfrak {c},l_{p}\left (Q\right )}$
,
$\mathcal {V}_{p^{-1}\mathfrak {c},l_{p}\left (Q\right ),\mathbb {C}_{p}}^{0}$
, and
$\mathcal {V}_{p^{-1} \mathfrak {c},l_{p}\left (Q\right ),\mathbb {C}_{p}}^{0}\left (\tfrac {1}{p+1}\right )$
of
$\mathcal {Y}_{p^{-1}\mathfrak {c},p,\mathbb {C}_{p}}$
, which are equal to the images of
$\mathcal {V}_{\mathfrak {c},Q}$
,
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}$
, and
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}} {}^{0}\left (\tfrac {1}{p+1}\right )$
by
$l_{p}$
, respectively. By Corollary 3.9, we have
$$ \begin{align*} q_{1}^{-1}\left(r^{-1}\left(\mathcal{V}_{p^{-1}\mathfrak{c},l_{p}\left(Q\right)}\right)\right)\subseteq q_{2}^{-1}\left(\mathcal{U}^{1}_{\mathfrak{c}}\right). \end{align*} $$
Taking the base extension, we also have
$$ \begin{align*} q_{1}^{-1}\left(r^{-1}\left(\mathcal{V}_{p^{-1}\mathfrak{c},l_{p}\left(Q\right),\mathbb{C}_{p}}^{0}\right)\right)\subseteq q_{1}^{-1}\left(r^{-1}\left(\mathcal{V}_{p^{-1}\mathfrak{c},l_{p}\left(Q\right),\mathbb{C}_{p}}\right)\right)\subseteq q_{2}^{-1}\left(\mathcal{U}^{1}_{\mathfrak{c},\mathbb{C}_{p}}\right). \end{align*} $$
Similarly, Lemma 3.10 shows
$r^{-1}\left (\mathcal {V}_{p^{-1}\mathfrak {c},l_{p}\left (Q\right ),\mathbb {C}_{p}}^{0}\left (\tfrac {1}{p+1}\right )\right )\subseteq \mathcal {U}^{1}_{p^{-1}\mathfrak {c},\mathbb {C}_{p}}$
. Since the weight
$\kappa _{1}$
is integral, we have a natural isomorphism
$\pi _{p}^{*}:q_{2}^{*}\left (\pi ^{1}\right )^{*}\Omega ^{\kappa _{1}} \to q_{1}^{*}\left (\pi ^{1}\right )^{*}\Omega ^{\kappa _{1}}$
over
$ \mathcal {Z}^{1}_{p^{-1}\mathfrak {c},p}$
. From these and the foregoing commutative diagrams, we see that the operator
$U_{p}$
extends to an operator
$$ \begin{align*} U_{Q}: H^{0}\left(\mathcal{U}^{1}_{\mathfrak{c},\mathbb{C}_{p}},\left(\pi^{1}\right)^{*}\Omega^{\kappa_{1}}\right)\to H^{0}\left(r^{-1}\left(\mathcal{V}_{\mathfrak{c},Q,\mathbb{C}_{p}}^{0}\right), \left(\pi^{1}\right)^{*}\Omega^{\kappa_{1}}\right) \end{align*} $$
which makes the following diagram commutative:
Step 5. Now suppose that
$f(O)$
is of infinite slope. Then
$$ \begin{align*}\left(U_{Q}g_{\mathfrak{c}}\right)\left\rvert_{r^{-1}\left(\mathcal{V}_{\mathfrak{c},Q,\mathbb{C}_{p}}^{0}\left(\tfrac{1}{p+1}\right)\right)}\right.=\left(h^{*}U_{p}f_{\mathfrak{c}}(O)\right)\left\rvert_{r^{-1}\left(\mathcal{V}_{\mathfrak{c},Q,\mathbb{C}_{p}}^{0}\left(\tfrac{1}{p+1}\right)\right)}\right.=0. \end{align*} $$
Since
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}$
is connected and r is finite and étale, the map r defines a surjection from each connected component of
$r^{-1} \left (\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}\right )$
to
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}$
. Since the admissible open subset
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}\left (\tfrac {1}{p+1}\right )$
is nonempty, we see that
$r^{-1}\left (\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}\left (\tfrac {1}{p+1}\right )\right )$
intersects every connected component of
$r^{-1}\left (\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}\right )$
. Thus Lemma 3.1 implies
$U_{Q}g_{\mathfrak {c}}=0$
. In particular, if the point
$[(A,\lambda ,\mathcal {L})]\in Y_{\mathfrak {c},p}\left (\mathcal {O}_{\bar {\mathbb {Q}}_{p}}\right )$
satisfies
$\mathrm {Hdg}_{\beta }(A)=p/(p+1)$
for any
$\beta \in \mathbb {B}_{F}$
, then for any
$\mathcal {O}_{F}$
-isomorphism
$m: \mathcal {D}_{F}^{-1}\otimes \mu _{p}\simeq \mathcal {L}_{K}$
we have
$$ \begin{align} \sum_{\mathcal{D}_{K}\cap \mathcal{L}_{K}=0} g_{\mathfrak{c}}\left(A/\mathcal{D}, p\bar{\lambda},\bar{m}\right)=0, \end{align} $$
where the sum is taken over the set of finite flat closed p-cyclic
$\mathcal {O}_{F}$
-subgroup schemes
$\mathcal {D}$
of
$A[p]$
satisfying
$\mathcal {D}_{K}\cap \mathcal {L}_{K}=0$
and
$\bar {\lambda }$
is the induced
$p\mathfrak {c}$
-polarization. We will omit
$p\bar {\lambda }$
from test objects as in the foregoing.
Lemma 5.2. For any p-cyclic
$\mathcal {O}_{F}$
-subgroup scheme
$\mathcal {H}$
of
$A[p]$
and any
$\mathcal {O}_{F}$
-isomorphism
$u:\mathcal {D}_{F}^{-1}\otimes \mu _{p}\to (A[p]/\mathcal {H})_{K}$
, we have
$g_{\mathfrak {c}}(A/\mathcal {H}, u)=0$
.
Proof. For any p-cyclic
$\mathcal {O}_{F}$
-subgroup scheme
$\mathcal {M}$
of
$A[p]$
, write
$\mathcal {M}=\bigoplus _{\mathfrak {p}\mid p} \mathcal {M}_{\mathfrak {p}}$
. Similarly, any
$\mathcal {O}_{F}$
-closed immersion
$m:\mathcal {D}_{F}^{-1} \otimes \mu _{p}\to A_{K}$
defines a closed immersion
$m_{\mathfrak {p}}: \mathcal {D}^{-1}_{F}/\mathfrak {p}\mathcal {D}^{-1}_{F} \otimes \mu _{p} \to A[\mathfrak {p}]_{K}$
for any
$\mathfrak {p}\mid p$
. By fixing a generator of the principal
$\mathcal {O}_{F}$
-module
$\mathcal {D}_{F}^{-1}/p\mathcal {D}_{F}^{-1}$
and a primitive pth root of unity in
$\bar {\mathbb {Q}}_{p}$
, we identify an
$\mathcal {O}_{F}$
-closed immersion
$m:\mathcal {D}^{-1} {}_{F}\otimes \mu _{p} \to A_{K}$
with an element of
$A[p]\left (\bar {\mathbb {Q}}_{p}\right )$
. Let
$\mathfrak {P}$
be the set of maximal ideals of
$\mathcal {O}_{F}$
dividing p. For any subset
$S\subseteq \mathfrak {P}$
, we put
$S^{c}=\mathfrak {P}\setminus S$
and
$$ \begin{align*} \mathcal{M}_{S}=\bigoplus_{\mathfrak{p}\in S}\mathcal{M}_{\mathfrak{p}},\qquad \mathcal{M}^{S}=\bigoplus_{\mathfrak{p}\in S^{c}} \mathcal{M}_{\mathfrak{p}},\qquad \mathcal{M}_{\emptyset}=\mathcal{M}^{\mathfrak{P}}=0. \end{align*} $$
We define
$m_{S}$
and
$m^{S}$
similarly. We write
$\mathrm {Im}(m)$
also as
$\langle m \rangle $
.
For any
$\mathfrak {p}\mid p$
, we fix nonzero elements
$e_{\mathfrak {p},1}\in \mathcal {H}_{\mathfrak {p}}\left (\bar {\mathbb {Q}}_{p}\right )$
and
$e_{\mathfrak {p},2}\in A[\mathfrak {p}]\left (\bar {\mathbb {Q}}_{p}\right )$
such that
$\left \{e_{\mathfrak {p},1},e_{\mathfrak {p},2} \right \}$
forms a basis of the
$\mathfrak {o}/\mathfrak {p}$
-module
$A[\mathfrak {p}]\left (\bar {\mathbb {Q}}_{p}\right )$
. Put
$I_{\mathfrak {p}}=\left \{e_{\mathfrak {p},1}, a_{\mathfrak {p}} e_{\mathfrak {p},1}+e_{\mathfrak {p},2}\mid a_{\mathfrak {p}}\in \mathfrak {o}/\mathfrak {p}\right \}$
and
$e_{S,i}=\left (e_{\mathfrak {p},i}\right )_{\mathfrak {p}\in S}$
for
$i=1,2$
. We claim that for any element
$m^{S}$
of
$\prod _{\mathfrak {p}\in S^{c}}I_{\mathfrak {p}}$
, we have
$$ \begin{align} \sum_{\mathcal{D}^{S}_{K}\cap \left\langle m^{S} \right\rangle=0}g_{\mathfrak{c}}\left(A/\left(\mathcal{H}_{S}\times \mathcal{D}^{S}\right),\overline{e_{S,2}\times m^{S}}\right)=0, \end{align} $$
where the sum is taken over the set of finite flat closed
$\left (\prod _{\mathfrak {p}\in S^{c}}\mathfrak {p}\right )$
-cyclic
$\mathcal {O}_{F}$
-subgroup schemes
$\mathcal {D}^{S}$
of A satisfying
$\mathcal {D}^{S}_{K}\cap \left \langle m^{S} \right \rangle =0$
.
To show the claim, we proceed by induction on
$\sharp S$
. The case of
$S=\emptyset $
is equation (5.1). Suppose that the claim holds for some
$S\neq \mathfrak {P}$
. Take
$\mathfrak {p}\in S^{c}$
and put
$S^{\prime }=S\cup \{\mathfrak {p}\}$
. Fix
$m^{S^{\prime }}\in \prod _{\mathfrak {q}\in (S^{\prime })^{c}}I_{\mathfrak {q}}$
. Taking the sum of equation (5.2) over the set
$\left \{m^{S}=m_{\mathfrak {p}} \times m^{S^{\prime }}\mid m_{\mathfrak {p}}\in I_{\mathfrak {p}}\right \}$
, we obtain
$$ \begin{align*} \sum_{m_{\mathfrak{p}}\in I_{\mathfrak{p}}}\sum_{\mathcal{D}_{\mathfrak{p},K}\cap \left\langle m_{\mathfrak{p}} \right\rangle=0} \sum_{\mathcal{D}^{S^{\prime}}_{K}\cap \left\langle m^{S^{\prime}} \right\rangle=0}g_{\mathfrak{c}}\left(A/\left(\mathcal{H}_{S}\times \mathcal{D}_{\mathfrak{p}}\times \mathcal{D}^{S^{\prime}}\right),\overline{e_{S,2}\times m_{\mathfrak{p}}\times m^{S^{\prime}}}\right)=0. \end{align*} $$
We compute terms in this sum for each
$\mathcal {D}_{\mathfrak {p}}$
.
-
• If
$\mathcal {D}_{\mathfrak {p}}\left (\bar {\mathbb {Q}}_{p}\right )=(\mathfrak {o}/\mathfrak {p})e_{\mathfrak {p},1}=\mathcal {H}_{\mathfrak {p}}\left (\bar {\mathbb {Q}}_{p}\right )$
and
$\mathcal {D}_{\mathfrak {p},K}\cap \left \langle m_{\mathfrak {p}} \right \rangle =0$
, then
$m_{\mathfrak {p}}=a_{\mathfrak {p}} e_{\mathfrak {p},1}+e_{\mathfrak {p},2}$
with some
$a_{\mathfrak {p}}\in \mathfrak {o}/\mathfrak {p}$
. In this case,
$\bar {m}_{\mathfrak {p}}$
is equal to the image
$\bar {e}_{\mathfrak {p},2}$
of
$e_{\mathfrak {p},2}$
. -
• If
$\mathcal {D}_{\mathfrak {p}}\left (\bar {\mathbb {Q}}_{p}\right )=(\mathfrak {o}/\mathfrak {p})\left (a_{\mathfrak {p}} e_{\mathfrak {p},1}+e_{\mathfrak {p},2}\right )$
and
$\mathcal {D}_{\mathfrak {p},K}\cap \left \langle m_{\mathfrak {p}} \right \rangle =0$
, then we have either
$m_{\mathfrak {p}}=e_{\mathfrak {p}, 1}$
or
$m_{\mathfrak {p}}=b_{\mathfrak {p}} e_{\mathfrak {p},1}+e_{\mathfrak {p},2}$
with some
$b_{\mathfrak {p}}\neq a_{\mathfrak {p}}\in \mathfrak {o}/\mathfrak {p}$
. In each case,
$\bar {m}_{\mathfrak {p}}$
is equal to the element
$\bar {e}_{\mathfrak {p},1}$
or
$\left (b_{\mathfrak {p}}-a_{\mathfrak {p}}\right )\bar {e}_{\mathfrak {p},1}$
.
Thus the sum of the terms in which
$\mathcal {D}_{\mathfrak {p}}$
s of the second case appear is equal to
$$ \begin{align*} p^{f_{\mathfrak{p}}}\sum_{\mathcal{D}^{S^{\prime}}_{K}\cap \left\langle m^{S^{\prime}} \right\rangle=0}\sum_{a_{\mathfrak{p}}\in \mathfrak{o}/\mathfrak{p}}g_{\mathfrak{c}}\left(A/\left(\mathcal{H}_{S}\times (\mathfrak{o}/\mathfrak{p})\left(a_{\mathfrak{p}} e_{\mathfrak{p},1}+e_{\mathfrak{p},2}\right)\times \mathcal{D}^{S^{\prime}}\right),\overline{e_{S,2}\times e_{\mathfrak{p},1}\times m^{S^{\prime}}}\right). \end{align*} $$
This equals
$$ \begin{align*} p^{f_{\mathfrak{p}}}\sum_{\mathcal{D}^{S}_{K}\cap \left\langle e_{\mathfrak{p},1}\times m^{S^{\prime}}\right\rangle=0}g_{\mathfrak{c}}\left(A/\left(\mathcal{H}_{S}\times \mathcal{D}^{S}\right), \overline{e_{S,2}\times e_{\mathfrak{p},1}\times m^{S^{\prime}}}\right), \end{align*} $$
which is zero by the induction hypothesis (5.2). What remains is the sum of the terms of
$\mathcal {D}_{\mathfrak {p}}$
s of the first case, which equals
$$ \begin{align*} p^{f_{\mathfrak{p}}}\sum_{\mathcal{D}^{S^{\prime}}_{K}\cap \left\langle m^{S^{\prime}} \right\rangle=0}g_{\mathfrak{c}}\left(A/\left(\mathcal{H}_{S^{\prime}}\times \mathcal{D}^{S^{\prime}}\right), \overline{e_{S^{\prime},2}\times m^{S^{\prime}}}\right)=0 \end{align*} $$
and the claim follows. Setting
$S=\mathfrak {P}$
, we obtain
$g_{\mathfrak {c}}\left (A/\mathcal {H}, \bar {e}_{\mathfrak {P},2}\right )=0$
. For any u as in the lemma, the map
$u_{\mathfrak {p}}$
corresponds to
$a_{\mathfrak {p}} \bar {e}_{\mathfrak {p},2}$
for some
$a_{\mathfrak {p}}\in (\mathfrak {o}/\mathfrak {p})^{\times }$
. Thus we have
$$ \begin{align*} g_{\mathfrak{c}}(A/\mathcal{H}, u)=g_{\mathfrak{c}}\left(A/\mathcal{H}, \bar{e}_{\mathfrak{P},2}\right)=0, \end{align*} $$
and the lemma follows.
Consider the admissible open subset of
$\mathcal {Y}_{\mathfrak {c},p}$
defined by
$$ \begin{align*} \left\{[(A,\lambda,\mathcal{H})]\mid \mathrm{Hdg}_{\beta}(A)=p/(p+1)\text{ for any }\beta\in\mathbb{B}_{F}\right\} \end{align*} $$
and let
$\mathcal {V}$
be a nonempty admissible affinoid open subset of it. Note that the map
$$ \begin{align*} W: \mathcal{Y}_{\mathfrak{c},p}\to \mathcal{Y}_{\mathfrak{c},p},\quad (A,\lambda,\mathcal{H})\mapsto \left(A/\mathcal{H},p\bar{\lambda},A[p]/\mathcal{H}\right) \end{align*} $$
is an isomorphism. By [Reference Hattori29, Proposition 6.1], we have
$r^{-1}(W(\mathcal {V}))\subseteq \mathcal {U}^{1}_{\mathfrak {c}}$
. Consider the base extensions
$W_{\mathbb {C}_{p}}: \mathcal {Y}_{\mathfrak {c},p, \mathbb {C}_{p}}\to \mathcal {Y}_{\mathfrak {c},p,\mathbb {C}_{p}}$
and
$\mathcal {V}_{\mathbb {C}_{p}}$
, where the latter is an admissible affinoid open subset of
$\mathcal {Y}_{\mathfrak {c},p,\mathbb {C}_{p}}$
. By Lemma 5.2,
$\pi ^{*}f_{\mathfrak {c}}(O)$
vanishes on the subset
$W(\mathcal {V})\left (\bar {\mathbb {Q}}_{p}\right )$
of the admissible affinoid open subset
$W_{\mathbb {C}_{p}}\left (\mathcal {V}_{\mathbb {C}_{p}}\right )=W(\mathcal {V})_{\mathbb {C}_{p}}$
.
Lemma 5.3. Let A be a reduced K-affinoid algebra. Put
$X=\mathrm {Sp}(A)$
,
$A_{\mathbb {C}_{p}}=A\hat {\otimes }_{K} \mathbb {C}_{p}$
, and
$X_{\mathbb {C}_{p}}=\mathrm {Sp}\left (A_{\mathbb {C}_{p}}\right )$
. We consider the set
$X\left (\bar {\mathbb {Q}}_{p}\right )$
as a subset of
$X_{\mathbb {C}_{p}}\left (\mathbb {C}_{p}\right )$
by the natural inclusion
$\bar {\mathbb {Q}}_{p}\to \mathbb {C}_{p}$
. Suppose that an element
$f\in A_{\mathbb {C}_{p}}$
satisfies
$f(x)=0$
for any
$x\in X\left (\bar {\mathbb {Q}} _{p}\right )$
. Then
$f=0$
.
Proof. For any positive rational number
$\varepsilon $
, we put
$$ \begin{align*} U_{\varepsilon}=\left\{x\in X_{\mathbb{C}_{p}}\mid \lvert f(x)\rvert\leq \varepsilon\right\}. \end{align*} $$
We can find an element
$f_{\varepsilon }\in A\otimes _{K} \bar {\mathbb {Q}}_{p}$
such that
$$ \begin{align*} \lvert(f-f_{\varepsilon})(x)\rvert\leq \varepsilon\text{ for any }x\in X_{\mathbb{C}_{p}}. \end{align*} $$
Then we have
$U_{\varepsilon }=\left \{x\in X_{\mathbb {C}_{p}}\mid \lvert f_{\varepsilon }(x)\rvert \leq \varepsilon \right \}$
. Take a finite extension
$L/K$
satisfying
$f_{\varepsilon }\in A_{L}:=A\otimes _{K} L$
. Put
$X_{L}=\mathrm {Sp}(A_{L})$
. The assumption implies
$X\left (\bar {\mathbb {Q}}_{p}\right )\subseteq U_{\varepsilon }$
, namely
$\lvert f_{\varepsilon }(x)\rvert \leq \varepsilon $
for any
$x\in X\left (\bar {\mathbb {Q}} _{p}\right )$
. This shows
$X_{L}=\{x\in X_{L}\mid \lvert f_{\varepsilon }(x)\rvert \leq \varepsilon \}$
. Since the formation of rational subsets is compatible with base extensions, we have
$X_{\mathbb {C}_{p}}=U_{\varepsilon }$
for any
$\varepsilon>0$
, which implies
$f(x)=0$
for any
$x\in X_{\mathbb {C}_{p}}$
. Since
$X_{\mathbb {C}_{p}}$
is reduced, we obtain
$f=0$
and the lemma follows.
Since the invertible sheaf
$\pi ^{*}\Omega ^{\kappa }$
is the base extension to
$\mathbb {C}_{p}$
of a similar invertible sheaf over K, it is trivialized by the base extension of an admissible affinoid covering over K. By Lemma 5.3, we have
$\pi ^{*}f_{\mathfrak {c}}(O)\rvert _{W(\mathcal {V})_{\mathbb {C}_{p}}}=0$
. Thus
$f_{\mathfrak {c}}(O)$
vanishes on the admissible open subset
$\pi \left (W(\mathcal {V})_{\mathbb {C}_{p}}\right )$
of
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\tfrac {1}{p+1}\right )_{\mathbb {C}_{p}}$
. By Lemma 3.4,
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\tfrac {1}{p+1}\right )_{\mathbb {C}_{p}}$
is connected. By Lemma 3.1, we obtain
$f_{\mathfrak {c}}(O)=0$
for any
$\mathfrak {c}$
, which contradicts the fact that
$f(O)$
is normalized. This concludes the proof of the theorem.
6 List of notation
Following is a list of the main notation, in order of definition.
§1:
$\mathcal {D}_{\mathbb {C}_{p}}$
, O,
$\mathcal {D}_{\mathbb {C}_{p}}^{\times }$
§3.1: F, g,
$\mathfrak {o}$
,
$\mathcal {D}_{F}$
,
$U_{N}$
,
$f_{\mathfrak {p}}$
, K, k, W,
$\pi _{K}$
, e,
$v_{p}$
,
$m_{L}^{\geq i}$
,
$\mathcal {O}_{L,i}$
,
$\mathscr {S}_{L,i}$
,
$\mathbb {B}_{F}$
,
$\mathbb {B}_{\mathfrak {p}}$
,
$F_{\mathbb {R}}^{*}$
,
$F_{\mathbb {R}}^{*,+}$
,
$\mathrm {Dec}(\mathfrak {a},\mathfrak {b})$
,
$\mathscr {C}(\mathfrak {a},\mathfrak {b})$
,
$\mathscr {D}(\mathfrak {c})$
§3.1.1:
$\omega _{\mathcal {G},\beta }$
,
$\deg _{\beta }$
,
$\mathrm {Hdg}_{\beta }$
§3.1.2:
$M(\mu _{N},\mathfrak {c})$
,
$A^{\mathrm {un}}$
,
$\sigma ^{\vee }$
,
$q^{m}$
,
$\hat {S}_{\sigma }$
,
$\hat {R}_{\sigma }$
,
$\hat {S}_{\mathscr {C}}$
,
$\hat {I}_{\sigma }$
,
$\bar {S}_{\sigma }$
,
$\bar {S}_{\sigma }^{0}$
,
$\hat {R}_{\sigma }^{0}$
,
$\mathfrak {X}_{\eta }$
,
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q)$
,
$\lambda _{\mathfrak {a},\mathfrak {b}}$
,
$\bar {M}(\mu _{N},\mathfrak {c})$
, D,
$\bar {A}^{\mathrm {un}}$
§3.1.3:
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})$
,
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})$
,
$\mathcal {M}(\mu _{N},\mathfrak {c})$
,
$\bar {\mathfrak {A}}^{\mathrm {un}}$
,
$\bar {\mathcal {A}}^{\mathrm {un}}$
,
$\omega _{\bar {A}^{\mathrm {un}}}$
,
$\omega _{\bar {A}^{\mathrm {un}},\beta }$
,
$\omega _{\bar {A}^{\mathrm {un}}}^{\kappa }$
,
$\omega _{\bar {\mathcal {A}}^{\mathrm {un}},\beta }$
,
$\omega _{\bar {\mathcal {A}}^{\mathrm {un}}}^{\kappa }$
,
$h_{\beta }$
,
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
,
$\mathcal {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
,
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
,
$\mathfrak {M}(\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
,
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v)$
,
$\bar {\mathcal {M}}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
,
$\bar {\mathfrak {M}}(\mu _{N},\mathfrak {c})(v)$
,
$\mathbf {NAdm}$
,
$\tau \left (\bar {P}\right )$
§3.1.4:
$\mathcal {C}_{n}$
,
$C_{n}$
,
$\mathrm {HT}_{\mathcal {G}}$
,
$\delta _{K}$
§3.2:
$Y_{\mathfrak {c},p}$
,
$\mathfrak {Y}_{\mathfrak {c},p}$
,
$\mathcal {Y}_{\mathfrak {c},p}$
,
$\mathcal {V}_{\mathfrak {c},Q}$
,
$\mathcal {V}_{\mathfrak {c},Q}\left (\frac {1}{p+1}\right )$
,
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}$
,
$\mathcal {V}_{\mathfrak {c},Q,\mathbb {C}_{p}}^{0}\left (\frac {1}{p+1}\right )$
§3.3.1:
$\mathbb {T}$
,
$\hat {\mathbb {T}}$
,
$\mathbb {T}^{0}_{w}$
,
$\mathcal {W}$
,
$\kappa ^{\mathrm {un}}$
,
$\mathcal {O}^{\circ }$
,
$\kappa ^{\mathcal {X}}$
,
$\delta _{n}$
,
$B_{n}$
,
$\bar {\mathcal {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
,
$\bar {\mathfrak {M}}(\Gamma _{1}(p^{n}),\mu _{N},\mathfrak {c})\left (\underline {v}\right )$
,
$\gamma _{w}$
,
$\mathfrak {IW}_{w}^{+}$
,
$\mathfrak {IW}_{w,\mathfrak {c}}^{+}\left (\underline {v}\right )$
,
$\mathcal {IW}_{w}^{+}$
,
$\mathcal {IW}_{w,\mathfrak {c}}^{+}\left (\underline {v}\right )$
,
$h_{w}$
,
$\pi _{w}$
,
$\mathbb {T}_{w}$
,
$\Omega ^{\kappa ^{\mathcal {U}}}$
,
$M\left (\mu _{N},\mathfrak {c},\kappa ^{\mathcal {U}}\right )\left (\underline {v}\right )$
,
$S\left (\mu _{N},\mathfrak {c},\kappa ^{\mathcal {U}}\right )\left (\underline {v}\right )$
,
$\mathbf {\Omega }^{\kappa ^{\mathcal {U}}}$
§3.3.2:
$\mathcal {W}^{G}$
,
$k(\nu ,w)$
,
$\Delta $
,
$F^{\times ,+,(p)}$
,
$\mathrm {Frac}(F)^{(p)}$
,
$M^{G}\left (\mu _{N},\mathfrak {c},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )\left (\underline {v}\right )$
,
$S^{G}\left (\mu _{N},\mathfrak {c},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )\left (\underline {v}\right )$
,
$L_{x}$
,
$M^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )\left (\underline {v}\right )$
,
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )\left (\underline {v}\right )$
,
$M^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v_{\mathrm {tot}})$
,
$S^{G}\left (\mu _{N},\left (\nu ^{\mathcal {U}},w^{\mathcal {U}}\right )\right )(v_{\mathrm {tot}})$
§3.3.3:
$\mathcal {Y}^{\prime }_{\mathfrak {c},\mathfrak {l}}\left (\underline {v}\right )$
,
$p_{1}$
,
$p_{2}$
,
$\pi _{\mathfrak {l}}$
,
$x_{\mathfrak {p}}$
,
$T_{\mathfrak {l}}$
,
$S_{\mathfrak {l}}$
,
$U_{\mathfrak {l}}$
,
$U_{p}$
,
$\mathbf {T}$
,
$v_{n}$
,
$\mathcal {E}$
§3.4:
$\pi _{w,\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\gamma _{w,\mathcal {O}_{\mathbb {C}_{p}}}$
,
$h_{n,\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\bar {M}^{*}(\mu _{N},\mathfrak {c})$
,
$\bar {\mathfrak {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
,
$\bar {\mathcal {M}}^{*}(\mu _{N},\mathfrak {c})(v_{\mathrm {tot}})$
,
$\rho $
,
$\rho ^{\mathrm {rig}}$
,
$\rho _{\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\rho ^{\mathrm {rig}}_{\mathbb {C}_{p}}$
§4:
$\left [\mathrm {Cl}^{+}(F)\right ]^{(p)}$
,
$f_{\mathfrak {c}}$
§4.1:
$\breve {S}_{\sigma }$
,
$\breve {R}_{\sigma }$
,
$\mathcal {O}_{L}\left [X_{\leq r},X^{\pm }_{>r}\right ]$
,
$\mathcal {O}_{L}\left \langle X_{\leq r},X^{\pm }_{>r}\right \rangle $
,
$L\left \langle X_{\leq r},X^{\pm }_{>r}\right \rangle $
,
$\breve {S}^{\mathrm {rig}}_{\sigma }$
,
$\breve {S}_{\mathscr {C}}$
,
$\breve {S}^{\mathrm {rig}}_{\mathscr {C}}$
,
$(-)^{\mathrm {rig}}_{/\mathbb {C}_{p}}$
,
$\breve {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\breve {S}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\breve {S}_{\mathscr {C},\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\tilde {S}_{\sigma }$
,
$\tilde {R}_{\sigma ,\mathcal {O}_{\mathbb {C}_{p}}}$
,
$\tau _{\mathfrak {a},\mathfrak {b}}$
,
$\mathcal {O}_{\mathcal {K}_{\sigma }}$
,
$f_{\mathfrak {c}}(q)$
,
$a_{\mathfrak {o},\mathfrak {c}^{-1}}(f,\xi )$
§4.2:
$\Lambda (\mathfrak {n})$
,
$\Phi (\mathfrak {n})$
,
$\mu _{\mathfrak {n}}$
,
$\zeta ^{\eta }$
§4.2.1:
$\hat {R}_{\eta ,\sigma }$
,
$\hat {R}^{0}_{\eta ,\sigma }$
,
$\breve {R}_{\eta ,\sigma }$
,
$q^{\eta }$
,
$\mathrm {Tate}_{\mathfrak {a},\mathfrak {b}}(q^{\eta })$
§4.3:
$\mathbf {M}(\mu _{N},\mathfrak {c},\kappa )(0)$
,
$\mathcal {S}_{\mathfrak {c}}$
§4.4:
$a_{\mathfrak {o},\mathfrak {o}}^{\mathcal {U}}(f,1)$
§5:
$\mathcal {U}$
,
$\mathcal {X}_{\mathfrak {c}}$
,
$\mathcal {X}_{\mathfrak {c}}(v^{\prime })$
, h,
$\mathcal {X}^{1}_{\mathfrak {c}}(v^{\prime })$
,
$\mathcal {Y}_{\mathfrak {c},p}(v^{\prime })$
, r,
$\mathcal {Y}^{1}_{\mathfrak {c},p}$
,
$\pi ^{1}$
,
$\mathcal {Y}^{1}_{\mathfrak {c},p}(v^{\prime })$
,
$\mathcal {U}^{1}_{\mathfrak {c}}$
,
$g_{\mathfrak {c}}$
,
$Z_{\mathfrak {c},p}$
,
$\mathcal {Z}_{\mathfrak {c},p}$
,
$q_{1}$
,
$q_{2}$
,
$\mathcal {Z}_{\mathfrak {c},p}(v^{\prime })$
,
$\Pi $
Acknowledgments
The author would like to thank Fabrizio Andreatta, Ruochuan Liu, and Vincent Pilloni for kindly answering his questions on their works. He is also grateful to Kevin Buzzard, Yoichi Mieda, Tomoki Mihara, Tadashi Ochiai, Vincent Pilloni, and Yuichiro Taguchi for stimulating comments on an earlier draft, and the anonymous referee for valuable comments. He also would like to thank Shu Sasaki for enlightening discussions on p-adic modular forms and encouragements.
This work was supported by JSPS KAKENHI grant numbers JP26400016, JP20K03545.
Competing Interests
None.
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