1 Introduction
The primary goal of this paper is to explain how (under mild technical hypotheses) the patching construction of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16], when applied to the group
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
, gives rise to the
$p$
-adic local Langlands correspondence for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
, as constructed in [Reference ColmezCol10b] and as further analyzed in [Reference PaškūnasPaš13] and [Reference Colmez, Dospinescu and PaškūnasCDP14]. As a by-product, we obtain a new proof of many cases of the local–global compatibility theorem of [Reference EmertonEme11] (and of some cases not treated there).
1.1 Background
We start by recalling the main results of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] and the role we expect them to play in the (hypothetical)
$p$
-adic local Langlands correspondence. Let
$F$
be a finite extension of
$\mathbb{Q}_{p}$
and let
$G_{F}$
be its absolute Galois group. One would like to have an analogue of the local Langlands correspondence for all finite-dimensional, continuous,
$p$
-adic representations of
$G_{F}$
. Let
$E$
be another finite extension of
$\mathbb{Q}_{p}$
, which will be our field of coefficients, assumed large enough, with ring of integers
${\mathcal{O}}$
, uniformizer
$\unicode[STIX]{x1D71B}$
and residue field
$\mathbb{F}$
. To a continuous Galois representation
$r:G_{F}\rightarrow \operatorname{GL}_{n}(E)$
one would like to attach an admissible unitary
$E$
-Banach space representation
$\unicode[STIX]{x1D6F1}(r)$
of
$G:=\operatorname{GL}_{n}(F)$
(or possibly a family of such Banach space representations). Ideally, such a construction should be compatible with deformations, should encode the classical local Langlands correspondence and should be compatible with a global
$p$
-adic correspondence, realized in the completed cohomology of locally symmetric spaces.
It is expected that the Banach spaces
$\unicode[STIX]{x1D6F1}(r)$
should encode the classical local Langlands correspondence in the following way: if
$r$
is potentially semistable with regular Hodge–Tate weights, then the subspace of locally algebraic vectors
$\unicode[STIX]{x1D6F1}(r)^{\text{l.alg}}$
in
$\unicode[STIX]{x1D6F1}(r)$
should be isomorphic to
$\unicode[STIX]{x1D70B}_{\text{sm}}(r)\otimes \unicode[STIX]{x1D70B}_{\text{alg}}(r)$
as a
$G$
-representation, where
$\unicode[STIX]{x1D70B}_{\text{sm}}(r)$
is the smooth representation of
$G$
corresponding via classical local Langlands to the Weil–Deligne representation obtained from
$r$
by Fontaine’s recipe, and
$\unicode[STIX]{x1D70B}_{\text{alg}}(r)$
is an algebraic representation of
$G$
, whose highest-weight vector is determined by the Hodge–Tate weights of
$r$
.
Example 1.2. If
$F=\mathbb{Q}_{p}$
,
$n=2$
and
$r$
is crystalline with Hodge–Tate weights
$a<b$
, then
$\unicode[STIX]{x1D70B}_{\text{sm}}(r)$
is a smooth unramified principal series representation, whose Satake parameters can be calculated in terms of the trace and determinant of Frobenius on
$D_{\text{cris}}(r)$
, and
$\unicode[STIX]{x1D70B}_{\text{alg}}(r)=\text{Sym}^{b-a-1}E^{2}\otimes \;\det ^{1-a}$
. (We note that in the literature different normalizations lead to different twists by a power of
$\det$
.)
Such a correspondence has been established in the case of
$n=2$
and
$F=\mathbb{Q}_{p}$
by the works of Breuil, Colmez and others; see [Reference BreuilBre08, Reference ColmezCol10a] as well as the introduction to [Reference ColmezCol10b]. Moreover, when
$n=2$
and
$F=\mathbb{Q}_{p}$
, this correspondence has been proved (in most cases) to satisfy local–global compatibility with the
$p$
-adically completed cohomology of modular curves; see [Reference EmertonEme11]. However, not much is known beyond this case. In [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] we have constructed a candidate for such a correspondence using the Taylor–Wiles–Kisin patching method, which has been traditionally employed to prove modularity lifting theorems for Galois representations. We now describe the end product of the paper [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16].
Let
$\bar{r}:G_{F}\rightarrow \operatorname{GL}_{n}(\mathbb{F})$
be a continuous representation and let
$R_{p}^{\Box }$
be its universal framed deformation ring. Under the assumption that
$p$
does not divide
$2n$
, we construct an
$R_{\infty }[G]$
-module
$M_{\infty }$
, which is finitely generated as a module over the completed group algebra
$R_{\infty }[[\operatorname{GL}_{n}({\mathcal{O}}_{F})]]$
, where
$R_{\infty }$
is a complete local noetherian
$R_{p}^{\Box }$
-algebra with residue field
$\mathbb{F}$
. If
$y\in \operatorname{Spec}R_{\infty }$
is an
$E$
-valued point, then
$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{y}:=\operatorname{Hom}_{{\mathcal{O}}}^{\text{cont}}(M_{\infty }\otimes _{R_{\infty },y}{\mathcal{O}},E)\end{eqnarray}$$
is an admissible unitary
$E$
-Banach space representation of
$G$
. The composition
$R_{p}^{\Box }\rightarrow R_{\infty }\overset{y}{\rightarrow }E$
defines an
$E$
-valued point
$x\in \operatorname{Spec}R_{p}^{\Box }$
and thus a continuous Galois representation
$r_{x}:G_{F}\rightarrow \operatorname{GL}_{n}(E)$
. We expect that the Banach space representation
$\unicode[STIX]{x1D6F1}_{y}$
depends only on
$x$
and that it should be related to
$r_{x}$
by the hypothetical
$p$
-adic Langlands correspondence; see [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16, § 6] for a detailed discussion. We show in [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16, Theorem 4.35] that if
$\unicode[STIX]{x1D70B}_{\text{sm}}(r_{x})$
is generic and
$x$
lies on an automorphic component of a potentially crystalline deformation ring of
$\bar{r}$
, then
$\unicode[STIX]{x1D6F1}_{y}^{\text{l.alg}}\cong \unicode[STIX]{x1D70B}_{\text{sm}}(r_{x})\otimes \unicode[STIX]{x1D70B}_{\text{alg}}(r_{x})$
, as expected; moreover, the points
$x$
such that
$\unicode[STIX]{x1D70B}_{\text{sm}}(r_{x})$
is generic are Zariski dense in every irreducible component of a potentially crystalline deformation ring. (It is expected that every irreducible component of a potentially crystalline deformation ring is automorphic; this expectation is motivated by the Fontaine–Mazur and Breuil–Mézard conjectures. However, it is intrinsic to our method that we would not be able to access these non-automorphic components even if they existed.)
However, there are many natural questions regarding our construction for
$\operatorname{GL}_{n}(F)$
that we cannot answer at the moment and that appear to be genuinely deep, as they are intertwined with questions about local–global compatibility for
$p$
-adically completed cohomology, with the Breuil–Mézard conjecture on the geometry of local deformation rings and with the Fontaine–Mazur conjecture for global Galois representations. For example, it is not clear that
$\unicode[STIX]{x1D6F1}_{y}$
depends only on
$x$
, it is not clear that
$\unicode[STIX]{x1D6F1}_{y}$
is non-zero for an arbitrary
$y$
and that furthermore
$\unicode[STIX]{x1D6F1}_{y}^{\text{l.alg}}$
is non-zero if
$r_{x}$
is potentially semistable of regular weight and it is not at all clear that
$M_{\infty }$
does not depend on the different choices made during the patching process.
1.3 The present paper
In this paper, we specialize the construction of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] to the case
$F=\mathbb{Q}_{p}$
and
$n=2$
(so that
$G:=\operatorname{GL}_{2}(\mathbb{Q}_{p})$
and
$K:=\operatorname{GL}_{2}(\mathbb{Z}_{p})$
from now on) to confirm our expectation that, firstly,
$M_{\infty }$
does not depend on any of the choices made during the patching process and, secondly, that it does recover the
$p$
-adic local Langlands correspondence as constructed by Colmez.
We achieve the first part without appealing to Colmez’s construction (which relies on the theory of
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules). The proof that
$M_{\infty }$
is uniquely determined highlights some key features of the
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
setting beyond the use of
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules: the classification of irreducible mod
$p$
representations of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
in terms of Serre weights and Hecke operators, and the fact that the Weil–Deligne representation and the Hodge–Tate weights determine a (irreducible) two-dimensional crystalline representation of
$G_{\mathbb{Q}_{p}}$
uniquely (up to isomorphism).
When combined with the results of [Reference PaškūnasPaš13] (which do rely on Colmez’s functor
$\check{\mathbf{V}}$
), we obtain that
$M_{\infty }$
realizes the
$p$
-adic Langlands correspondence as constructed by Colmez.
We also obtain a new proof of local–global compatibility, which helps clarify the relationship between different perspectives and approaches to
$p$
-adic local Langlands.
1.4 Arithmetic actions
In the body of the paper we restrict the representations
$\bar{r}$
we consider by assuming that they have only scalar endomorphisms, so that
$\text{End}_{G_{\mathbb{Q}_{p}}}(\bar{r})=\mathbb{F}$
and that
$\bar{r}\not \cong \big(\!\begin{smallmatrix}\unicode[STIX]{x1D714} & \ast \\ 0 & 1\end{smallmatrix}\!\big)\otimes \unicode[STIX]{x1D712}$
for any character
$\unicode[STIX]{x1D712}:G_{\mathbb{Q}_{p}}\rightarrow \mathbb{F}^{\times }$
. For simplicity, let us assume in this introduction that
$\bar{r}$
is irreducible and let
$R_{p}$
be its universal deformation ring. Then
$R_{p}^{\Box }$
is formally smooth over
$R_{p}$
. Moreover (as
$F=\mathbb{Q}_{p}$
and
$n=2$
), we may also assume that
$R_{\infty }$
is formally smooth over
$R_{p}^{\Box }$
and thus over
$R_{p}$
.
The following definition is meant to axiomatize the key properties of the patched module
$M_{\infty }$
.
Definition 1.5. Let
$d$
be a non-negative integer, let
$R_{\infty }:=R_{p}[[x_{1},\ldots ,x_{d}]]$
and let
$M$
be a non-zero
$R_{\infty }[G]$
-module. We say that the action of
$R_{\infty }$
on
$M$
is arithmetic if the following conditions hold:
- (AA1)
$M$
is a finitely generated module over the completed group algebra
$R_{\infty }[[K]]$
;- (AA2)
- $M$
is projective in the category of pseudocompact
${\mathcal{O}}[[K]]$
-modules;
- (AA3)
for each pair of integers
$a<b$
, the action of
$R_{\infty }$
on $$\begin{eqnarray}M(\unicode[STIX]{x1D70E}^{\circ }):=\operatorname{Hom}_{{\mathcal{O}}[[K]]}^{\text{cont}}(M,(\unicode[STIX]{x1D70E}^{\circ })^{d})^{d}\end{eqnarray}$$
factors through the action of
$R_{\infty }(\unicode[STIX]{x1D70E}):=R_{p}(\unicode[STIX]{x1D70E})[[x_{1},\ldots ,x_{d}]]$
. Here
$R_{p}(\unicode[STIX]{x1D70E})$
is the quotient of
$R_{p}$
constructed by Kisin, which parameterizes crystalline representations with Hodge–Tate weights
$(a,b)$
,
$\unicode[STIX]{x1D70E}^{\circ }$
is a
$K$
-invariant
${\mathcal{O}}$
-lattice in
$\unicode[STIX]{x1D70E}:=\text{Sym}^{b-a-1}E^{2}\otimes \det ^{1-a}$
and
$(\ast )^{d}:=\operatorname{Hom}_{{\mathcal{O}}}^{\text{cont}}(\ast ,{\mathcal{O}})$
denotes the Schikhof dual.Moreover,
$M(\unicode[STIX]{x1D70E}^{\circ })$
is maximal Cohen–Macaulay over
$R_{\infty }(\unicode[STIX]{x1D70E})$
and the
$R_{\infty }(\unicode[STIX]{x1D70E})[1/p]$
-module
$M(\unicode[STIX]{x1D70E}^{\circ })[1/p]$
is locally free of rank
$1$
over its support;- (AA4)
for each
$\unicode[STIX]{x1D70E}$
as above and each maximal ideal
$y$
of
$R_{\infty }[1/p]$
in the support of
$M(\unicode[STIX]{x1D70E}^{\circ })$
, there is a non-zero
$G$
-equivariant map $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\text{sm}}(r_{x})\otimes \unicode[STIX]{x1D70B}_{\text{alg}}(r_{x})\rightarrow \unicode[STIX]{x1D6F1}_{y}^{\text{l.alg}},\end{eqnarray}$$
where
$x$
is the image of
$y$
in
$\operatorname{Spec}R_{p}$
.
The last condition says that
$M$
encodes the classical local Langlands correspondence. This is what motivated us to call such actions arithmetic. (In fact in the main body of the paper we use a reformulation of condition (AA4); see § 3.1 and Remark 3.3.) To motivate (AA3), we note for the sake of the reader familiar with Kisin’s proof of the Fontaine–Mazur conjecture [Reference KisinKis09] that the modules
$M(\unicode[STIX]{x1D70E}^{\circ })$
are analogues of the patched modules denoted by
$M_{\infty }$
in [Reference KisinKis09], except that Kisin patches algebraic automorphic forms for definite quaternion algebras and in this paper we will ultimately be making use of patching arguments for algebraic automorphic forms on forms of
$U(2)$
.
1.6 Uniqueness of
$M_{\infty }$
As already mentioned, the patched module
$M_{\infty }$
of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] carries an arithmetic action of
$R_{\infty }$
for some
$d$
. In order to prove that
$M_{\infty }$
is uniquely determined, it is enough to show that for any given
$d$
, any
$R_{\infty }[G]$
-module
$M$
with an arithmetic action of
$R_{\infty }$
is uniquely determined. The following is our main result, which for simplicity we state under the assumption that
$\bar{r}$
is irreducible.
Theorem 1.7. Let
$M$
be an
$R_{\infty }[G]$
-module with an arithmetic action of
$R_{\infty }$
.
(i) If
$\unicode[STIX]{x1D70B}$
is any irreducible
$G$
-subrepresentation of the Pontryagin dual
$M^{\vee }$
of
$M$
, then
$\unicode[STIX]{x1D70B}$
is isomorphic to the representation of
$G$
associated to
$\bar{r}$
by the mod
$p$
local Langlands correspondence for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
.(ii) Let
$\unicode[STIX]{x1D70B}{\hookrightarrow}J$
be an injective envelope of the above
$\unicode[STIX]{x1D70B}$
in the category of smooth locally admissible representations of
$G$
on
${\mathcal{O}}$
-torsion modules. Let
$\widetilde{P}$
be the Pontryagin dual of
$J$
. Then
$\widetilde{P}$
carries a unique arithmetic action of
$R_{p}$
and, moreover, as$$\begin{eqnarray}M\cong \widetilde{P}\;\operatorname{⊗̂}_{R_{p}}\,R_{\infty }\end{eqnarray}$$
$R_{\infty }[G]$
-modules.
The theorem completely characterizes modules with an arithmetic action and shows that
$M_{\infty }$
does not depend on the choices made in the patching process. A further consequence is that the Banach space
$\unicode[STIX]{x1D6F1}_{y}$
depends only on the image of
$y$
in
$\operatorname{Spec}R_{p}$
, as expected.
Let us sketch the proof of Theorem 1.7 assuming for simplicity that
$d=0$
. The first step is to show that
$M^{\vee }$
is an injective object in the category of smooth locally admissible representations of
$G$
on
${\mathcal{O}}$
-torsion modules and that its
$G$
-socle is isomorphic to
$\unicode[STIX]{x1D70B}$
. This is done by computing
$\operatorname{Hom}_{G}(\unicode[STIX]{x1D70B}^{\prime },M^{\vee })$
and showing that
$\text{Ext}_{G}^{1}(\unicode[STIX]{x1D70B}^{\prime },M_{\infty }^{\vee })$
vanishes for all irreducible
$\mathbb{F}$
-representations
$\unicode[STIX]{x1D70B}^{\prime }$
of
$G$
; see Proposition 4.2 and Theorem 4.15. The arguments here use the foundational results of Barthel and Livné [Reference Barthel and LivnéBL94] and Breuil [Reference BreuilBre03a] on the classification of irreducible mod
$p$
representations of
$G$
, arguments related to the weight part of Serre’s conjecture and the fact that the rings
$R_{p}(\unicode[STIX]{x1D70E})$
are formally smooth over
${\mathcal{O}}$
, whenever
$\unicode[STIX]{x1D70E}$
is of the form
$\text{Sym}^{b-a-1}E^{2}\otimes \;\det ^{1-a}$
with
$1\leqslant b-a\leqslant p$
.
This first step allows us to conclude that
$M^{\vee }$
is an injective envelope of
$\unicode[STIX]{x1D70B}$
, which depends only on
$\bar{r}$
. Since injective envelopes are unique up to isomorphism, we conclude that any two modules with an arithmetic action of
$R_{p}$
are isomorphic as
$G$
-representations. Therefore, it remains to show that any two arithmetic actions of
$R_{p}$
on
$\widetilde{P}$
coincide. As
$R_{p}$
is
${\mathcal{O}}$
-torsion free, it is enough to show that two such actions induce the same action on the unitary
$E$
-Banach space
$\unicode[STIX]{x1D6F1}:=\operatorname{Hom}_{{\mathcal{O}}}^{\text{cont}}(M,E)$
. Since
$M$
is a projective
${\mathcal{O}}[[K]]$
-module by (AA2), one may show using the ‘capture’ arguments that appear in [Reference Colmez, Dospinescu and PaškūnasCDP14, § 2.4] and [Reference EmertonEme11, Proposition 5.4.1] that the subspace of
$K$
-algebraic vectors in
$\unicode[STIX]{x1D6F1}$
is dense. Since the actions of
$R_{p}$
on
$\unicode[STIX]{x1D6F1}$
are continuous, it is enough to show that they agree on this dense subspace. Since the subspace of
$K$
-algebraic vectors is semisimple as a
$K$
-representation, it is enough to show that the two actions agree on
$\unicode[STIX]{x1D70E}$
-isotypic subspaces in
$\unicode[STIX]{x1D6F1}$
for all irreducible algebraic
$K$
-representations
$\unicode[STIX]{x1D70E}$
. These are precisely the representations
$\unicode[STIX]{x1D70E}$
in axiom (AA3). Taking duals one more time, we are left with showing that any two arithmetic actions induce the same action of
$R_{p}$
on
$M(\unicode[STIX]{x1D70E}^{\circ })[1/p]$
for all
$\unicode[STIX]{x1D70E}$
as above.
At this point we use another special feature of two-dimensional crystalline representations of
$G_{\mathbb{Q}_{p}}$
: the associated Weil–Deligne representation together with Hodge–Tate weights determine a two-dimensional crystalline representation of
$G_{\mathbb{Q}_{p}}$
up to isomorphism. Using this fact and axioms (AA3) and (AA4) for the arithmetic action, we show that the action of the Hecke algebra
${\mathcal{H}}(\unicode[STIX]{x1D70E}):=\text{End}_{G}(\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E})$
on
$M(\unicode[STIX]{x1D70E}^{\circ })[1/p]$
completely determines the action of
$R_{p}(\unicode[STIX]{x1D70E})$
on
$M(\unicode[STIX]{x1D70E}^{\circ })[1/p]$
; see the proof of Theorem 4.30 as well as the key Proposition 2.13. Since the action of
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
on
$M(\unicode[STIX]{x1D70E}^{\circ })[1/p]$
depends only on the
$G$
-module structure of
$M$
, we are able to conclude that the two arithmetic actions are the same. The reduction from the case when
$d$
is arbitrary to the case when
$d=0$
is carried out in § 4.16.
Remark 1.8. As we have already remarked, the arguments up to this point make no use of
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules. Indeed, the proof of Theorem 1.7 does not use them. One of the objectives of this project was to find out how much of the
$p$
-adic Langlands for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
correspondence can one recover from the patched module
$M_{\infty }$
without using Colmez’s functors, as these constructions are not available for groups other than
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
, while our patched module
$M_{\infty }$
is. Along the same lines, in § 5 we show that to a large extent we can recover a fundamental theorem of Berger and Breuil [Reference Berger and BreuilBB10] on the uniqueness of unitary completions of locally algebraic principal series without making use of
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules; see Theorem 5.1 and Remark 5.3.
Remark 1.9. As already explained, Theorem 1.7 implies that
$\unicode[STIX]{x1D6F1}_{y}$
depends only on the image of
$y$
in
$\operatorname{Spec}R_{p}$
. However, we are still not able to deduce using only our methods that
$\unicode[STIX]{x1D6F1}_{y}$
is non-zero for an arbitrary
$y\in \operatorname{m - Spec}R_{\infty }[1/p]$
. Since
$M_{\infty }$
is not a finitely generated module over
$R_{\infty }$
, theoretically it could happen that
$\unicode[STIX]{x1D6F1}_{y}\neq 0$
for a dense subset of
$\operatorname{m - Spec}R_{\infty }[1/p]$
, but
$\unicode[STIX]{x1D6F1}_{y}=0$
at all other maximal ideals. We can only prove that this pathological situation does not occur after combining Theorem 1.7 with the results of [Reference PaškūnasPaš13].
In § 6 we relate the arithmetic action of
$R_{p}$
on
$\widetilde{P}$
to the results of [Reference PaškūnasPaš13], where an action of
$R_{p}$
on an injective envelope of
$\unicode[STIX]{x1D70B}$
in the subcategory of representations with a fixed central character is constructed using Colmez’s functor; see Theorem 6.18. Then by appealing to the results of [Reference PaškūnasPaš13] we show that
$\unicode[STIX]{x1D6F1}_{y}$
and
$r_{x}$
correspond to each other under the
$p$
-adic Langlands correspondence as defined by Colmez [Reference ColmezCol10b] for all
$y\in \operatorname{m - Spec}R_{\infty }[1/p]$
, where
$x$
denotes the image of
$y$
in
$\operatorname{m - Spec}R_{p}[1/p]$
.
It follows from the construction of
$M_{\infty }$
that after quotienting out by a certain ideal of
$R_{\infty }$
we obtain a dual of completed cohomology; see [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16, Corollary 2.11]. This property combined with Theorem 1.7 and with the results in § 6 enables us to obtain a new proof of local–global compatibility as in [Reference EmertonEme11] as well as obtaining a genuinely new result, when
$\overline{\unicode[STIX]{x1D70C}}|_{G_{\mathbb{Q}_{p}}}$
is isomorphic to
$\big(\!\begin{smallmatrix}1 & \ast \\ 0 & \unicode[STIX]{x1D714}\end{smallmatrix}\!\big)\otimes \unicode[STIX]{x1D712}$
, where
$\unicode[STIX]{x1D714}$
is the cyclotomic character modulo
$p$
. (See Remark 7.7.)
1.10 Prospects for generalization
Since our primary goal in this paper is to build some new connections between various existing ideas related to the
$p$
-adic Langlands program, we have not striven for maximal generality, and we expect that some of our hypotheses on
$\bar{r}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{2}(\overline{\mathbb{F}}_{p})$
could be relaxed. In particular, it should be possible to prove results when
$p=2$
by using results of Thorne [Reference ThorneTho17] to redo the patching in [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16]. It may also be possible to extend our results to cover more general
$\bar{r}$
(recall that we assume that
$\bar{r}$
has only scalar endomorphisms, and that it is not a twist of an extension of the trivial character by the mod
$p$
cyclotomic character). In § 6.27 we discuss the particular case where
$\bar{r}$
has scalar semisimplification; as this discussion (and the arguments of [Reference PaškūnasPaš13]) show, while it may well be possible to generalize our arguments, they will necessarily be considerably more involved in cases where
$\bar{r}$
does not satisfy the hypotheses that we have imposed.
Since the patching construction in [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] applies equally well to the case of
$\operatorname{GL}_{2}(F)$
for any finite extension
$F/\mathbb{Q}_{p}$
, or indeed to
$\operatorname{GL}_{n}(F)$
, it is natural to ask whether any of our arguments can be extended to such cases (where there is at present no construction of a
$p$
-adic local Langlands correspondence). As explained in Remark 3.2, the natural analogues of our axioms (AA1)–(AA4) hold, even in the generality of
$\operatorname{GL}_{n}(F)$
. Unfortunately, the prospects for proving analogues of our main theorems are less rosy, as it seems that none of the main inputs to our arguments will hold. Indeed, already for the case of
$\operatorname{GL}_{2}(\mathbb{Q}_{p^{2}})$
there is no analogue available of the classification in [Reference BreuilBre03a] of the irreducible
$\mathbb{F}$
-representations of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
, and it is clear from the results of [Reference Breuil and PaškūnasBP12] that any such classification would be much more complicated.
Furthermore, beyond the case of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
it is no longer the case that crystalline representations are (essentially) determined by their underlying Weil–Deligne representations, so there is no possibility of deducing that a
$p$
-adic correspondence is uniquely determined by the classical correspondence in the way that we do here, and no hope that an analogue of the results of [Reference Berger and BreuilBB10] could hold. Finally, it is possible to use the constructions of [Reference PaškūnasPaš04] to show that for
$\operatorname{GL}_{2}(\mathbb{Q}_{p^{2}})$
the patched module
$M_{\infty }$
is not a projective
$\operatorname{GL}_{2}(\mathbb{Q}_{p^{2}})$
-module.
1.11 Outline of the paper
In § 2 we recall some well-known results about Hecke algebras and crystalline deformation rings for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
. The main result in this section is Proposition 2.15, which describes the crystalline deformation rings corresponding to Serre weights as completions of the corresponding Hecke algebras. In § 3 we explain our axioms for a module with an arithmetic action, and show how the results of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16] produce patched modules
$M_{\infty }$
satisfying these axioms.
Section 4 proves that the axioms determine
$M_{\infty }$
(essentially) uniquely, giving a new construction of the
$p$
-adic local Langlands correspondence for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
. It begins by showing that
$M_{\infty }$
is a projective
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
-module (Theorem 4.15), before making a category-theoretic argument that allows us to ‘factor out’ the patching variables (Proposition 4.22). We then use the ‘capture’ machinery to complete the proof.
In § 5 we explain how our results can be used to give a new proof (not making use of
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules) that certain locally algebraic principal series representations admit at most one unitary completion. Section 6 combines our results with those of [Reference PaškūnasPaš13] to show that our construction is compatible with Colmez’s correspondence, and as a by-product extends some results of [Reference PaškūnasPaš13] to a situation where the central character is not fixed.
Finally, in § 7 we explain how our results give a new proof of the second author’s local–global compatibility theorem, and briefly explain how such results can be extended to quaternion algebras over totally real fields (Remark 7.8).
1.12 Notation
We fix an odd prime
$p$
, an algebraic closure
$\overline{\mathbb{Q}}_{p}$
of
$\mathbb{Q}_{p}$
and a finite extension
$E/\mathbb{Q}_{p}$
in
$\overline{\mathbb{Q}}_{p}$
, which will be our coefficient field. We write
${\mathcal{O}}={\mathcal{O}}_{E}$
for the ring of integers in
$E$
,
$\unicode[STIX]{x1D71B}=\unicode[STIX]{x1D71B}_{E}$
for a uniformizer and
$\mathbb{F}:={\mathcal{O}}/\unicode[STIX]{x1D71B}$
for the residue field. We will assume without comment that
$E$
and
$\mathbb{F}$
are sufficiently large, and in particular that if we are working with representations of the absolute Galois group of a
$p$
-adic field
$K$
, then the images of all embeddings
$K{\hookrightarrow}\overline{\mathbb{Q}}_{p}$
are contained in
$E$
.
1.12.1 Galois-theoretic notation
If
$K$
is a field, we let
$G_{K}$
denote its absolute Galois group. Let
$\unicode[STIX]{x1D700}$
denote the
$p$
-adic cyclotomic character and
$\overline{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D714}$
the mod
$p$
cyclotomic character. If
$K$
is a finite extension of
$\mathbb{Q}_{p}$
for some
$p$
, we write
$I_{K}$
for the inertia subgroup of
$G_{K}$
. If
$R$
is a local ring, we write
$\mathfrak{m}_{R}$
for the maximal ideal of
$R$
. If
$F$
is a number field and
$v$
is a finite place of
$F$
, then we let
$\operatorname{Frob}_{v}$
denote a geometric Frobenius element of
$G_{F_{v}}$
.
If
$K/\mathbb{Q}_{p}$
is a finite extension, we write
$\operatorname{Art}_{K}:K^{\times }\stackrel{{\sim}}{\longrightarrow }W_{K}^{\text{ab}}$
for the Artin map normalized to send uniformizers to geometric Frobenius elements. To avoid cluttering up the notation, we will use
$\operatorname{Art}_{\mathbb{Q}_{p}}$
to regard characters of
$\mathbb{Q}_{p}^{\times }$
,
$\mathbb{Z}_{p}^{\times }$
as characters of
$G_{\mathbb{Q}_{p}}$
,
$I_{\mathbb{Q}_{p}}$
, respectively, without explicitly mentioning
$\operatorname{Art}_{\mathbb{Q}_{p}}$
when we do so.
If
$K$
is a
$p$
-adic field and
$\unicode[STIX]{x1D70C}$
a de Rham representation of
$G_{K}$
over
$E$
and if
$\unicode[STIX]{x1D70F}:K{\hookrightarrow}E$
, then we will write
$\operatorname{HT}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70C})$
for the multiset of Hodge–Tate numbers of
$\unicode[STIX]{x1D70C}$
with respect to
$\unicode[STIX]{x1D70F}$
. By definition, the multiset
$\operatorname{HT}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D70C})$
contains
$i$
with multiplicity
$\dim _{E}(\unicode[STIX]{x1D70C}\otimes _{\unicode[STIX]{x1D70F},K}\widehat{\overline{K}}(i))^{G_{K}}$
. Thus for example
$\operatorname{HT}_{\unicode[STIX]{x1D70F}}(\unicode[STIX]{x1D700})=\{-1\}$
. If
$\unicode[STIX]{x1D70C}$
is moreover crystalline, then we have the associated filtered
$\unicode[STIX]{x1D711}$
-module
$D_{\operatorname{cris}}(\unicode[STIX]{x1D70C}):=(\unicode[STIX]{x1D70C}\otimes _{\mathbb{Q}_{p}}B_{\text{cris}})^{G_{K}}$
, where
$B_{\text{cris}}$
is Fontaine’s crystalline period ring.
1.12.2 Local Langlands correspondence
Let
$n\in \mathbb{Z}_{{\geqslant}1}$
, let
$K$
be a finite extension of
$\mathbb{Q}_{p}$
and let
$\operatorname{rec}$
denote the local Langlands correspondence from isomorphism classes of irreducible smooth representations of
$\operatorname{GL}_{n}(K)$
over
$\mathbb{C}$
to isomorphism classes of
$n$
-dimensional Frobenius semisimple Weil–Deligne representations of
$W_{K}$
defined in [Reference Harris and TaylorHT01]. Fix an isomorphism
$\imath :\overline{\mathbb{Q}}_{p}\rightarrow \mathbb{C}$
. We define the local Langlands correspondence
$\operatorname{rec}_{p}$
over
$\overline{\mathbb{Q}}_{p}$
by
$\imath \circ \operatorname{rec}_{p}=\operatorname{rec}\circ \imath$
. Then
$r_{p}(\unicode[STIX]{x1D70B}):=\operatorname{rec}_{p}(\unicode[STIX]{x1D70B}\otimes |\det |^{(1-n)/2})$
is independent of the choice of
$\imath$
. In this paper we are mostly concerned with the case that
$n=2$
and
$K=\mathbb{Q}_{p}$
.
1.12.3 Notation for duals
If
$A$
is a topological
${\mathcal{O}}$
-module, we write
$A^{\vee }:=\operatorname{Hom}_{{\mathcal{O}}}^{\text{cont}}(A,E/{\mathcal{O}})$
for the Pontryagin dual of
$A$
. We apply this to
${\mathcal{O}}$
-modules that are either discrete or profinite, so that the usual formalism of Pontryagin duality applies.
If
$A$
is a pseudocompact
${\mathcal{O}}$
-torsion-free
${\mathcal{O}}$
-module, we write
$A^{d}:=\operatorname{Hom}_{{\mathcal{O}}}^{\text{cont}}(A,{\mathcal{O}})$
for its Schikhof dual.
If
$F$
is a free module of finite rank over a ring
$R$
, then we write
$F^{\ast }:=\operatorname{Hom}_{R}(F,R)$
to denote its
$R$
-linear dual, which is again a free
$R$
-module of the same rank over
$R$
as
$F$
.
If
$R$
is a commutative
${\mathcal{O}}$
-algebra, and if
$A$
is an
$R$
-module that is pseudocompact and
${\mathcal{O}}$
-torsion free as an
${\mathcal{O}}$
-module, then we may form its Schikhof dual
$A^{d}$
, which has a natural
$R$
-module structure via the transpose action, extending its
${\mathcal{O}}$
-module structure. If
$F$
is a finite-rank free
$R$
-module, then
$A\otimes _{R}F$
is again an
$R$
-module that is pseudocompact as an
${\mathcal{O}}$
-module (if
$F$
has rank
$n$
, then it is non-canonically isomorphic to a direct sum of
$n$
copies of
$A$
) and there is a canonical isomorphism of
$R$
-modules
$(A\otimes _{R}F)^{d}\stackrel{{\sim}}{\longrightarrow }A^{d}\otimes _{R}F^{\ast }.$
1.12.4 Group-theoretic notation
Throughout the paper we write
$G=\operatorname{GL}_{2}(\mathbb{Q}_{p})$
and
$K=\operatorname{GL}_{2}(\mathbb{Z}_{p})$
, and let
$Z=Z(G)$
denote the centre of
$G$
. We also let
$B$
denote the Borel subgroup of
$G$
consisting of upper-triangular matrices, and
$T$
denote the diagonal torus contained in
$B$
.
If
$\unicode[STIX]{x1D712}:T\rightarrow E^{\times }$
is a continuous character, then we define the continuous induction
$(\operatorname{Ind}_{B}^{G}\unicode[STIX]{x1D712})_{\text{cont}}$
to be the
$E$
-vector space of continuous functions
$f:G\rightarrow E$
satisfying the condition
$f(bg)=\unicode[STIX]{x1D712}(b)f(g)$
for all
$b\in B$
and
$g\in G$
; it forms a
$G$
-representation with respect to the right regular action. If
$\unicode[STIX]{x1D712}$
is in fact a smooth character, then we may also form the smooth induction
$(\operatorname{Ind}_{B}^{G}\unicode[STIX]{x1D712})_{\text{sm}}$
; this is the
$E$
-subspace of
$(\operatorname{Ind}_{B}^{G}\unicode[STIX]{x1D712})_{\text{cont}}$
consisting of smooth functions, and is a
$G$
-subrepresentation of the continuous induction.
If
$\unicode[STIX]{x1D712}_{1}$
and
$\unicode[STIX]{x1D712}_{2}$
are continuous characters of
$\mathbb{Q}_{p}^{\times }$
, then the character
$\unicode[STIX]{x1D712}_{1}\otimes \unicode[STIX]{x1D712}_{2}:T\rightarrow E^{\times }$
is defined via
$\big(\!\begin{smallmatrix}a & 0\\ 0 & d\end{smallmatrix}\!\big)\mapsto \unicode[STIX]{x1D712}_{1}(a)\unicode[STIX]{x1D712}_{2}(d)$
. Any continuous
$E$
-valued character
$\unicode[STIX]{x1D712}$
of
$T$
is of this form, and
$\unicode[STIX]{x1D712}$
is smooth if and only if
$\unicode[STIX]{x1D712}_{1}$
and
$\unicode[STIX]{x1D712}_{2}$
are.
2 Galois deformation rings and Hecke algebras
2.1 Galois deformation rings
Recall that we assume throughout the paper that
$p$
is an odd prime. Fix a continuous representation
$\bar{r}:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{2}(\mathbb{F})$
, where as before
$\mathbb{F}/\mathbb{F}_{p}$
is a finite extension. Possibly enlarging
$\mathbb{F}$
, we fix a sufficiently large extension
$E/\mathbb{Q}_{p}$
with ring of integers
${\mathcal{O}}$
and residue field
$\mathbb{F}$
.
We will make the following assumption from now on.
Assumption 2.2. Assume that
$\text{End}_{G_{\mathbb{Q}_{p}}}(\bar{r})=\mathbb{F}$
and that
$\bar{r}\not \cong \big(\!\begin{smallmatrix}\unicode[STIX]{x1D714} & \ast \\ 0 & 1\end{smallmatrix}\!\big)\otimes \unicode[STIX]{x1D712}$
for any character
$\unicode[STIX]{x1D712}:G_{\mathbb{Q}_{p}}\rightarrow \mathbb{F}^{\times }$
.
In particular this assumption implies that
$\bar{r}$
has a universal deformation
${\mathcal{O}}$
-algebra
$R_{p}$
, and that either
$\bar{r}$
is (absolutely) irreducible or that
$\bar{r}$
is a non-split extension of characters.
We begin by recalling the relationship between crystalline deformation rings of
$\bar{r}$
and the representation theory of
$G:=\operatorname{GL}_{2}(\mathbb{Q}_{p})$
and
$K:=\operatorname{GL}_{2}(\mathbb{Z}_{p})$
. Given a pair of integers
$a\in \mathbb{Z}$
and
$b\in \mathbb{Z}_{{\geqslant}0}$
, we let
$\unicode[STIX]{x1D70E}_{a,b}$
be the absolutely irreducible
$E$
-representation
$\det ^{a}\otimes \;\text{Sym}^{b}E^{2}$
of
$K$
. Note that this is just the algebraic representation of highest weight
$(a+b,a)$
with respect to the Borel subgroup given by the upper-triangular matrices in
$G$
.
We say that a representation
$r:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{2}(\overline{\mathbb{Q}}_{p})$
is crystalline of Hodge type
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{a,b}$
if it is crystalline with Hodge–Tate weights
$(1-a,-a-b)$
;Footnote
1
we write
$R_{p}(\unicode[STIX]{x1D70E})$
for the reduced,
$p$
-torsion-free quotient of
$R_{p}$
corresponding to crystalline deformations of Hodge type
$\unicode[STIX]{x1D70E}$
.
2.3 The morphism from the Hecke algebra to the deformation ring
We briefly recall some results from [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16, § 4], specialized to the case of crystalline representations of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$
.
Set
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{a,b}$
and let
${\mathcal{H}}(\unicode[STIX]{x1D70E}):=\text{End}_{G}(\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E})$
. The action of
$K$
on
$\unicode[STIX]{x1D70E}$
extends to the action of
$G$
. This gives rise to the isomorphism of
$G$
-representations:
$$\begin{eqnarray}(\operatorname{c - Ind}_{K}^{G}\mathbf{1})\otimes \unicode[STIX]{x1D70E}\cong \operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E},\quad f\otimes v\mapsto [g\mapsto f(g)\unicode[STIX]{x1D70E}(g)v].\end{eqnarray}$$
The map
$$\begin{eqnarray}{\mathcal{H}}(\unicode[STIX]{x1D7D9})\rightarrow {\mathcal{H}}(\unicode[STIX]{x1D70E}),\quad \unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D719}\otimes \text{id}_{\unicode[STIX]{x1D70E}}\end{eqnarray}$$
is an isomorphism of
$E$
-algebras by [Reference Schneider and TeitelbaumST06, Lemma 1.4]. Using the above isomorphism we will identify elements of
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
with
$E$
-valued
$K$
-bi-invariant functions on
$G$
, supported on finitely many double cosets.
Proposition 2.5. Let
$S\in {\mathcal{H}}(\unicode[STIX]{x1D70E})$
be the function supported on the double coset of
$\big(\!\begin{smallmatrix}p & 0\\ 0 & p\end{smallmatrix}\!\big)$
, with value
$p^{2a+b}$
at
$\big(\!\begin{smallmatrix}p & 0\\ 0 & p\end{smallmatrix}\!\big)$
, and let
$T\in {\mathcal{H}}(\unicode[STIX]{x1D70E})$
be the function supported on the double cosetFootnote
2
of
$\big(\!\begin{smallmatrix}p & 0\\ 0 & 1\end{smallmatrix}\!\big)$
, with value
$p^{a+b}$
at
$\big(\!\begin{smallmatrix}p & 0\\ 0 & 1\end{smallmatrix}\!\big)$
. Then
${\mathcal{H}}(\unicode[STIX]{x1D70E})=E[S^{\pm 1},T]$
as an
$E$
-algebra.
Proof. This is immediate from (2.4) and the Satake isomorphism. ◻
Let
$r,s$
be integers with
$r<s$
and let
$t,d\in E$
, with
$d\in E^{\times }$
. We let
$D:=D(r,s,t,d)$
be the two-dimensional filtered
$\unicode[STIX]{x1D711}$
-module that has
$e_{1}$
,
$e_{2}$
as a basis of its underlying
$E$
-vector space, has its
$E$
-linear Frobenius endomorphism
$\unicode[STIX]{x1D711}$
given by
$$\begin{eqnarray}\unicode[STIX]{x1D711}(e_{1})=e_{2},\quad \unicode[STIX]{x1D711}(e_{2})=-de_{1}+te_{2}\end{eqnarray}$$
and has its Hodge filtration given by
$$\begin{eqnarray}\operatorname{Fil}^{i}D=D\quad \text{if }i\leqslant r,\quad \operatorname{Fil}^{i}D=Ee_{1}\quad \text{if }r+1\leqslant i\leqslant s\quad \text{and}\quad \operatorname{Fil}^{i}D=0\quad \text{if }i>s.\end{eqnarray}$$
We note that
$t$
is the trace and
$d$
is the determinant of
$\unicode[STIX]{x1D711}$
on
$D$
, and both are therefore determined uniquely by
$D$
. The same construction works if
$E$
is replaced with an
$E$
-algebra
$A$
. We will still write
$D(r,s,t,d)$
for the resulting
$\unicode[STIX]{x1D711}$
-module with
$A$
-coefficients if the coefficient algebra is clear from the context.
Lemma 2.6. If
$V$
is an indecomposable two-dimensional crystalline representation of
$G_{\mathbb{Q}_{p}}$
over
$E$
with distinct Hodge–Tate weights
$(s,r)$
, then there exists a unique pair
$(t,d)\in E\times E^{\times }$
such that
$D_{\operatorname{cris}}(V)\cong D(r,s,t,d)$
. Moreover,
$v_{p}(d)=r+s$
and
$v_{p}(t)\geqslant r$
.
Proof. This is well known, and is a straightforward computation using the fact that
$D_{\operatorname{cris}}(V)$
is weakly admissible. For the sake of completeness, we sketch the proof; the key fact one employs is that
$V\mapsto D_{\operatorname{cris}}(V)$
is a fully faithful embedding of the category of crystalline representations of
$G_{\mathbb{Q}_{p}}$
into the category of weakly admissible filtered
$\unicode[STIX]{x1D711}$
-modules. (Indeed, it induces an equivalence between these two categories, but that more difficult fact is not needed for this computation.) We choose
$e_{1}$
to be a basis for
$\operatorname{Fil}^{s}D_{\operatorname{cris}}(V)$
; the assumption that
$V$
is indecomposable implies that
$\operatorname{Fil}^{s}D_{\operatorname{cris}}(V)$
is not stable under
$\unicode[STIX]{x1D711},$
and so if we write
$e_{2}:=\unicode[STIX]{x1D711}(e_{1})$
then
$e_{1},e_{2}$
is a basis for
$D_{\operatorname{cris}}(V)$
, and
$\unicode[STIX]{x1D711}$
has a matrix of the required form for a uniquely determined
$t$
and
$d$
. The asserted relations between
$v_{p}(t)$
,
$v_{p}(d)$
,
$r$
and
$s$
follow from the weak admissibility of
$D_{\operatorname{cris}}(V)$
.◻
In fact, it will be helpful to state a generalization of the previous result to the context of finite-dimensional
$E$
-algebras. (Note that the definition of
$D(r,s,t,d)$
extends naturally to the case when
$t$
and
$d$
are taken to lie in such a finite-dimensional algebra.)
Lemma 2.7. If
$A$
is an Artinian local
$E$
-algebra with residue field
$E^{\prime }$
, and if
$V_{A}$
is a crystalline representation of rank
$2$
over
$A$
whose associated residual representation
$V_{E^{\prime }}:=E^{\prime }\otimes _{A}V_{A}$
is indecomposable with distinct Hodge–Tate weights
$(s,r)$
, and if
$D_{\operatorname{cris}}(V_{A})$
denotes the filtered
$\unicode[STIX]{x1D711}$
-module associated to
$V_{A}$
, then there exists a unique pair
$(t,d)\in A\times A^{\times }$
such that
$D_{\operatorname{cris}}(V_{A})\cong D(r,s,t,d)$
.
Proof. Choose a basis
$\overline{e}_{1}$
for
$\operatorname{Fil}^{s}D_{\operatorname{cris}}(V_{E^{\prime }})$
, and choose
$e_{1}\in \operatorname{Fil}^{s}D_{\operatorname{cris}}(V_{A})$
lifting
$\overline{e}_{1}$
. By Nakayama’s lemma,
$e_{1}$
generates
$\operatorname{Fil}^{s}D_{\operatorname{cris}}(V_{A})$
and, by considering the length of
$\operatorname{Fil}^{s}D_{\operatorname{cris}}(V_{A})$
as an
$E$
-vector space, we see that
$\operatorname{Fil}^{s}D_{\operatorname{cris}}(V_{A})$
is a free
$A$
-module of rank
$1$
.
Let
$e_{2}=\unicode[STIX]{x1D719}(e_{1})$
and write
$\overline{e}_{2}$
for the image of
$e_{2}$
in
$D_{\operatorname{cris}}(V_{E^{\prime }})$
. As in the proof of Lemma 2.6,
$\overline{e}_{1},\overline{e}_{2}$
is a basis of
$D_{\operatorname{cris}}(V_{E^{\prime }})$
, and thus by another application of Nakayama’s lemma,
$e_{1},e_{2}$
are an
$A$
-basis of
$D_{\operatorname{cris}}(V_{A})$
. The matrix of
$\unicode[STIX]{x1D719}$
in this basis is evidently of the required form.◻
Corollary 2.8. If
$V$
is an indecomposable two-dimensional crystalline representation of
$G_{\mathbb{Q}_{p}}$
over
$E$
with distinct Hodge–Tate weights
$(s,r)$
, for which
$D_{\operatorname{cris}}(V)\cong D(r,s,t_{0},d_{0})$
, then the formal crystalline deformation ring of
$V$
is naturally isomorphic to
$E[[t-t_{0},d-d_{0}]].$
Proof. This is immediate from Lemma 2.7 and the fact that
$V\mapsto D_{\operatorname{cris}}(V)$
is an equivalence of categories.◻
Suppose that
$\bar{r}$
has a crystalline lift of Hodge type
$\unicode[STIX]{x1D70E}$
. By [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16, Theorem 4.1], there is a natural
$E$
-algebra homomorphism
$\unicode[STIX]{x1D702}:{\mathcal{H}}(\unicode[STIX]{x1D70E})\rightarrow R_{p}(\unicode[STIX]{x1D70E})[1/p]$
interpolating a normalized local Langlands correspondence
$r_{p}$
(introduced in § 1.12). In order to characterize this map, one considers the composite
$$\begin{eqnarray}{\mathcal{H}}(\unicode[STIX]{x1D70E})\{R_{p}(\unicode[STIX]{x1D70E})[1/p]{\hookrightarrow}(R_{p}(\unicode[STIX]{x1D70E}))^{\text{an}},\end{eqnarray}$$
where
$(R_{p}(\unicode[STIX]{x1D70E}))^{\text{an}}$
denotes the ring of rigid analytic functions on the rigid analytic generic fibre of
$\operatorname{Spf}R_{p}(\unicode[STIX]{x1D70E})$
. Over
$(R_{p}(\unicode[STIX]{x1D70E}))^{\text{an}}$
, we may consider the universal filtered
$\unicode[STIX]{x1D711}$
-module, and the underlying universal Weil group representation (given by forgetting the filtration). The trace and determinant of Frobenius on this representation are certain elements of
$(R_{p}(\unicode[STIX]{x1D70E}))^{\text{an}}$
(which in fact lie in
$R_{p}(\unicode[STIX]{x1D70E})[1/p]$
), and
$\unicode[STIX]{x1D702}$
is characterized by the fact that it identifies appropriately chosen generators of
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
with these universal trace and determinant.
It is straightforward to give explicit formulas for these generators of
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
, but we have found it interesting (in part with an eye to making arguments in more general contexts) to also derive the facts that we need without using such explicit formulas.
Regarding explicit formulas, we have the following result.
Proposition 2.9. The elements
$\unicode[STIX]{x1D702}(S),\unicode[STIX]{x1D702}(T)\in R_{p}(\unicode[STIX]{x1D70E})[1/p]$
are characterized by the following property: if
$x:R_{p}(\unicode[STIX]{x1D70E})[1/p]\rightarrow \overline{\mathbb{Q}}_{p}$
is an
$E$
-algebra morphism, and
$V_{x}$
is the corresponding two-dimensional
$\overline{\mathbb{Q}}_{p}$
-representation of
$G_{\mathbb{Q}_{p}}$
, then
$x(\unicode[STIX]{x1D702}(T))=p^{a+b}t$
and
$x(\unicode[STIX]{x1D702}(S))=p^{2a+b-1}d$
, where
$t$
,
$d$
are respectively the trace and the determinant of
$\unicode[STIX]{x1D711}$
on
$D_{\operatorname{cris}}(V_{x})$
.
Proof. Lemma 2.7 implies that there are uniquely determined
$t,d\in \overline{\mathbb{Q}}_{p}$
such that
$$\begin{eqnarray}D_{\operatorname{cris}}(V_{x})\cong D(r,s,t,d),\end{eqnarray}$$
where
$r=-a-b$
and
$s=1-a$
. The Weil–Deligne representation associated to
$D(r,s,t,d)$
is an unramified two-dimensional representation of
$W_{\mathbb{Q}_{p}}$
, on which the geometric Frobenius
$\operatorname{Frob}_{p}$
acts by the matrix of crystalline Frobenius on
$D(r,s,t,d)$
, which is
$\big(\!\begin{smallmatrix}0 & -d\\ 1 & t\end{smallmatrix}\!\big)$
. Thus,
$$\begin{eqnarray}\text{WD}(D_{\operatorname{cris}}(V_{x}))=\operatorname{rec}_{p}(\unicode[STIX]{x1D712}_{1})\oplus \operatorname{rec}_{p}(\unicode[STIX]{x1D712}_{2}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D712}_{1},\unicode[STIX]{x1D712}_{2}:\mathbb{Q}_{p}^{\times }\rightarrow \overline{\mathbb{Q}}_{p}^{\times }$
are unramified characters such that
$\unicode[STIX]{x1D712}_{1}(p)+\unicode[STIX]{x1D712}_{2}(p)=t$
and
$\unicode[STIX]{x1D712}_{1}(p)\unicode[STIX]{x1D712}_{2}(p)=d$
.
If
$\unicode[STIX]{x1D70B}=(\operatorname{Ind}_{B}^{G}|\centerdot |\unicode[STIX]{x1D712}_{1}\otimes \unicode[STIX]{x1D712}_{2})_{\text{sm}}$
, then
$\unicode[STIX]{x1D70B}\otimes |\text{det}|^{-1/2}\cong \unicode[STIX]{x1D704}_{B}^{G}(\unicode[STIX]{x1D712}_{1}\otimes \unicode[STIX]{x1D712}_{2})$
, where
$\unicode[STIX]{x1D704}_{B}^{G}$
denotes smooth normalized parabolic induction. Then
$$\begin{eqnarray}r_{p}(\unicode[STIX]{x1D70B})=\operatorname{rec}_{p}(\unicode[STIX]{x1D704}_{B}^{G}(\unicode[STIX]{x1D712}_{1}\otimes \unicode[STIX]{x1D712}_{2}))=\operatorname{rec}_{p}(\unicode[STIX]{x1D712}_{1})\oplus \operatorname{rec}_{p}(\unicode[STIX]{x1D712}_{2}).\end{eqnarray}$$
The action of
${\mathcal{H}}(\mathbf{1})$
on
$\unicode[STIX]{x1D70B}^{K}$
is given by sending
$[K\big(\!\begin{smallmatrix}p & 0\\ 0 & 1\end{smallmatrix}\!\big)K]$
to
$p|p|\unicode[STIX]{x1D712}_{1}(p)+\unicode[STIX]{x1D712}_{2}(p)=t$
and
$[K\big(\!\begin{smallmatrix}p & 0\\ 0 & p\end{smallmatrix}\!\big)K]$
to
$|p|\unicode[STIX]{x1D712}_{1}(p)\unicode[STIX]{x1D712}_{2}(p)=p^{-1}d$
. By [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEGGPS16, Theorem 4.1] and the fact that the evident isomorphism between
$\unicode[STIX]{x1D70B}^{K}=\operatorname{Hom}_{K}(\mathbf{1},\mathbf{1}\otimes \unicode[STIX]{x1D70B})$
and
$\operatorname{Hom}_{K}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}\otimes \unicode[STIX]{x1D70B})$
is equivariant with respect to the actions by
${\mathcal{H}}(\mathbf{1})$
and
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
via the isomorphism (2.4), we see that
$$\begin{eqnarray}x(\unicode[STIX]{x1D702}(T))=p^{-r}t=p^{a+b}t,\quad x(\unicode[STIX]{x1D702}(S))=p^{-r-s}d=p^{2a+b-1}d.\end{eqnarray}$$
Since
$R_{p}(\unicode[STIX]{x1D70E})[1/p]$
is a reduced Jacobson ring, the formulas determine
$\unicode[STIX]{x1D702}(T)$
and
$\unicode[STIX]{x1D702}(S)$
uniquely.◻
Corollary 2.11.
$\unicode[STIX]{x1D702}(S)$
and
$\unicode[STIX]{x1D702}(T)$
are contained in the normalization of
$R_{p}(\unicode[STIX]{x1D70E})$
in
$R_{p}(\unicode[STIX]{x1D70E})[1/p]$
.
Proof. It follows from (2.10) and Lemma 2.6 that for all closed points
$x:R_{p}(\unicode[STIX]{x1D70E})[1/p]\rightarrow \overline{\mathbb{Q}}_{p}$
, we have
$x(\unicode[STIX]{x1D702}(S)),x(\unicode[STIX]{x1D702}(T))\in \overline{\mathbb{Z}}_{p}$
. The result follows from [Reference de JongdeJ95, Proposition 7.3.6].◻
A key fact that we will use, which is special to our context of two-dimensional crystalline representations of
$G_{\mathbb{Q}_{p}}$
, is that the morphism
$$\begin{eqnarray}(\operatorname{Spf}R(\unicode[STIX]{x1D70E}))^{\text{an}}\rightarrow (\operatorname{Spec}{\mathcal{H}}(\unicode[STIX]{x1D70E}))^{\text{an}}\end{eqnarray}$$
induced by
$\unicode[STIX]{x1D702}$
is an open immersion of rigid analytic spaces, where the superscript ‘an’ signifies the associated rigid analytic space. We prove this statement (in its infinitesimal form) in the following result.
Lemma 2.12. Let
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{a,b}$
, with
$a\in \mathbb{Z}$
and
$b\in \mathbb{Z}_{{\geqslant}0}$
. Then
$$\begin{eqnarray}\dim _{\unicode[STIX]{x1D705}(y)}\unicode[STIX]{x1D705}(y)\otimes _{{\mathcal{H}}(\unicode[STIX]{x1D70E})}R_{p}(\unicode[STIX]{x1D70E})[1/p]\leqslant 1,\quad \forall y\in \operatorname{m - Spec}{\mathcal{H}}(\unicode[STIX]{x1D70E}).\end{eqnarray}$$
Proof. Let us assume that
$A:=\unicode[STIX]{x1D705}(y)\otimes _{{\mathcal{H}}(\unicode[STIX]{x1D70E})}R_{p}(\unicode[STIX]{x1D70E})[1/p]$
is non-zero. If
$x,x^{\prime }\in \operatorname{m - Spec}A$
, then the Frobenii on
$D_{\operatorname{cris}}(V_{x})$
and
$D_{\operatorname{cris}}(V_{x^{\prime }})$
will have the same trace and determinant (since, by Proposition 2.9, these are determined by the images of
$T$
and
$S$
in
$\unicode[STIX]{x1D705}(y)$
); denote them by
$t$
and
$d$
. It follows from Lemma 2.6 that
$D_{\operatorname{cris}}(V_{x})\cong D_{\operatorname{cris}}(V_{x^{\prime }})$
and hence
$x=x^{\prime }$
. Since
$D(r,s,t,d)$
can be constructed over
$\unicode[STIX]{x1D705}(y)$
(as
$t$
and
$d$
lie in
$\unicode[STIX]{x1D705}(y)$
), so can
$V_{x}$
and thus
$\unicode[STIX]{x1D705}(x)=\unicode[STIX]{x1D705}(y)$
. To complete the proof of the lemma, it is enough to show that the map
$\mathfrak{m}_{y}\rightarrow \mathfrak{m}_{x}/\mathfrak{m}_{x}^{2}$
is surjective. Since we know that
$R_{p}(\unicode[STIX]{x1D70E})[1/p]$
is a regular ring of dimension
$2$
by [Reference KisinKis08, Theorem 3.3.8], it is enough to construct a two-dimensional family of deformations of
$D_{\operatorname{cris}}(V_{x})$
to the ring of dual numbers
$\unicode[STIX]{x1D705}(y)[\unicode[STIX]{x1D716}]$
, which induces a non-trivial deformation of the images of
$S,T$
. That this is possible is immediate from Corollary 2.8 and Proposition 2.9.◻
Proposition 2.13. Let
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{a,b}$
, with
$a\in \mathbb{Z}$
and
$b\in \mathbb{Z}_{{\geqslant}0}$
. Let
$y\in \operatorname{m - Spec}{\mathcal{H}}(\unicode[STIX]{x1D70E})$
be the image of
$x\in \operatorname{m - Spec}R_{p}(\unicode[STIX]{x1D70E})[1/p]$
under the morphism induced by
$\unicode[STIX]{x1D702}:{\mathcal{H}}(\unicode[STIX]{x1D70E})\rightarrow R_{p}(\unicode[STIX]{x1D70E})[1/p]$
. Then
$\unicode[STIX]{x1D702}$
induces an isomorphism of completions:
$$\begin{eqnarray}\widehat{{\mathcal{H}}(\unicode[STIX]{x1D70E})}_{\mathfrak{m}_{y}}\overset{\cong }{\longrightarrow }\widehat{R_{p}(\unicode[STIX]{x1D70E})[1/p]}_{\mathfrak{m}_{x}}.\end{eqnarray}$$
Proof. This can be proved by explicit computation, taking into account Corollary 2.8 and Proposition 2.9.
We can also deduce it in more pure thought manner as follows: since
${\mathcal{H}}(\unicode[STIX]{x1D70E})\cong E[T,S^{\pm 1}]$
by Proposition 2.5 and
$R_{p}(\unicode[STIX]{x1D70E})[1/p]$
is a regular ring of dimension
$2$
as in the preceding proof, both completions are regular rings of dimension
$2$
. It follows from Lemma 2.12 that
$\unicode[STIX]{x1D705}(y)=\unicode[STIX]{x1D705}(x)$
and the map induces a surjection on tangent spaces. Hence, the map is an isomorphism.◻
If
$0\leqslant b\leqslant p-1$
, then
$\unicode[STIX]{x1D70E}_{a,b}$
has a unique (up to homothety)
$K$
-invariant lattice
$\unicode[STIX]{x1D70E}_{a,b}^{\circ }$
, which is isomorphic to
$\det ^{a}\otimes \;\text{Sym}^{b}{\mathcal{O}}^{2}$
as a
$K$
-representation. We let
$\overline{\unicode[STIX]{x1D70E}}_{a,b}$
be its reduction modulo
$\unicode[STIX]{x1D71B}$
. Then
$\overline{\unicode[STIX]{x1D70E}}_{a,b}$
is the absolutely irreducible
$\mathbb{F}$
-representation
$\det ^{a}\otimes \;\text{Sym}^{b}\mathbb{F}^{2}$
of
$\operatorname{GL}_{2}(\mathbb{F}_{p})$
; note that every (absolutely) irreducible
$\mathbb{F}$
-representation of
$\operatorname{GL}_{2}(\mathbb{F}_{p})$
is of this form for some uniquely determined
$a,b$
with
$0\leqslant a<p-1$
. We refer to such representations as Serre weights.
If
$\overline{\unicode[STIX]{x1D70E}}=\overline{\unicode[STIX]{x1D70E}}_{a,b}$
is a Serre weight with the property that
$\bar{r}$
has a lift
$r:G_{\mathbb{Q}_{p}}\rightarrow \operatorname{GL}_{2}(\overline{\mathbb{Z}}_{p})$
that is crystalline of Hodge type
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{a,b}$
, then we say that
$\overline{\unicode[STIX]{x1D70E}}$
is a Serre weight of
$\bar{r}$
.
Again we consider
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{a,b}$
, with any
$a\in \mathbb{Z}$
and
$b\in \mathbb{Z}_{{\geqslant}0}$
. Let
$$\begin{eqnarray}\unicode[STIX]{x1D70E}^{\circ }:=\det \!^{a}\otimes \;\text{Sym}^{b}{\mathcal{O}}^{2},\quad \overline{\unicode[STIX]{x1D70E}}:=\det \!^{a}\otimes \;\text{Sym}^{b}\mathbb{F}^{2},\end{eqnarray}$$
so that
$\unicode[STIX]{x1D70E}^{\circ }/\unicode[STIX]{x1D71B}=\overline{\unicode[STIX]{x1D70E}}$
. We let
$$\begin{eqnarray}{\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ }):=\text{End}_{G}(\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ }),\quad {\mathcal{H}}(\overline{\unicode[STIX]{x1D70E}}):=\text{End}_{G}(\operatorname{c - Ind}_{K}^{G}\overline{\unicode[STIX]{x1D70E}}).\end{eqnarray}$$
Note that
${\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })$
is
$p$
-torsion free, since
$\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ }$
is.
Lemma 2.14.
(1) For any
$\unicode[STIX]{x1D70E}$
, there are a natural isomorphism
${\mathcal{H}}(\unicode[STIX]{x1D70E})\cong {\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })[1/p]$
and a natural inclusion
${\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })/\unicode[STIX]{x1D71B}{\hookrightarrow}{\mathcal{H}}(\overline{\unicode[STIX]{x1D70E}}).$
Furthermore, the
${\mathcal{O}}$
-subalgebra
${\mathcal{O}}[S^{\pm 1},T]$
of
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
is contained in
${\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })$
.(2) If, in addition,
$\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{a,b}$
, with
$0\leqslant b\leqslant p-1$
, then
${\mathcal{O}}[S^{\pm 1},T]={\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })$
and there is a natural isomorphism
${\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })/\unicode[STIX]{x1D71B}\cong {\mathcal{H}}(\overline{\unicode[STIX]{x1D70E}}).$
Proof. The isomorphism of (1) follows immediately from the fact that
$\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ }$
is a finitely generated
${\mathcal{O}}[G]$
-module. To see the claimed inclusion, apply
$\operatorname{Hom}_{G}(\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ },-)$
to the exact sequence
$$\begin{eqnarray}0\rightarrow \operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ }\overset{\unicode[STIX]{x1D71B}}{\rightarrow }\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ }\rightarrow \operatorname{c - Ind}_{K}^{G}\overline{\unicode[STIX]{x1D70E}}\rightarrow 0\end{eqnarray}$$
so as to obtain an injective map
$$\begin{eqnarray}{\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })/\unicode[STIX]{x1D71B}{\hookrightarrow}\operatorname{Hom}_{G}(\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ },\operatorname{c - Ind}_{K}^{G}\overline{\unicode[STIX]{x1D70E}})\cong {\mathcal{H}}(\overline{\unicode[STIX]{x1D70E}}).\end{eqnarray}$$
To see the final claim of (1), we recall that from (2.4) and Frobenius reciprocity we have natural isomorphisms
$$\begin{eqnarray}{\mathcal{H}}(\unicode[STIX]{x1D7D9})\simeq {\mathcal{H}}(\unicode[STIX]{x1D70E})\simeq \operatorname{Hom}_{K}(\unicode[STIX]{x1D70E},\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E});\end{eqnarray}$$
the image of
$\unicode[STIX]{x1D719}\in {\mathcal{H}}(\unicode[STIX]{x1D7D9})$
under the composite map sends
$v\in \unicode[STIX]{x1D70E}$
to the function
$g\mapsto \unicode[STIX]{x1D719}(g^{-1})\unicode[STIX]{x1D70E}(g)v$
. A direct computation of the actions of
$S,T$
on the standard basis of
$\unicode[STIX]{x1D70E}^{\circ }$
then verifies that
$S^{\pm 1}$
and
$T$
lie in the
${\mathcal{O}}$
-submodule
$\operatorname{Hom}_{K}(\unicode[STIX]{x1D70E}^{\circ },\operatorname{c - Ind}_{K}^{G}\unicode[STIX]{x1D70E}^{\circ })$
of
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
.
To prove (2), we note that it follows from [Reference BreuilBre03b, § 2] and [Reference Barthel and LivnéBL94] that the composite
$\mathbb{F}[S^{\pm 1},T]\rightarrow {\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })/\unicode[STIX]{x1D71B}\rightarrow {\mathcal{H}}(\overline{\unicode[STIX]{x1D70E}})$
is an isomorphism. Since the second of these maps is injective, by (1), we conclude that each of these maps is in fact an isomorphism, confirming the second claim of (2). Furthermore, this shows that the inclusion
${\mathcal{O}}[S^{\pm 1},T]{\hookrightarrow}{\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })$
of (1) becomes an isomorphism both after reducing modulo
$\unicode[STIX]{x1D71B}$
as well as after inverting
$\unicode[STIX]{x1D71B}$
(because
${\mathcal{H}}(\unicode[STIX]{x1D70E})$
is generated by
$S^{\pm 1}$
and
$T$
by Proposition 2.5). Thus, it is an isomorphism, completing the proof of (2).◻
The following lemma is well known, but for lack of a convenient reference we sketch a proof.
Lemma 2.15. Assume that
$\bar{r}$
satisfies Assumption 2.2. Then
$\bar{r}$
has at most two Serre weights. Furthermore, if we let
$\overline{\unicode[STIX]{x1D70E}}=\overline{\unicode[STIX]{x1D70E}}_{a,b}$
be a Serre weight of
$\bar{r}$
, then the following hold.
(i) The deformation ring
$R_{p}(\unicode[STIX]{x1D70E})$
is formally smooth of relative dimension
$2$
over
${\mathcal{O}}$
.(ii) The morphism of
$E$
-algebras
$\unicode[STIX]{x1D702}:{\mathcal{H}}(\unicode[STIX]{x1D70E})\rightarrow R_{p}(\unicode[STIX]{x1D70E})[1/p]$
induces a morphism of
${\mathcal{O}}$
-algebras
${\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })\rightarrow R_{p}(\unicode[STIX]{x1D70E})$
.(iii) The character
$\unicode[STIX]{x1D714}^{1-2a-b}\det \bar{r}$
is unramified and, if we let:–
$\unicode[STIX]{x1D707}=(\unicode[STIX]{x1D714}^{1-2a-b}\det \bar{r})(\operatorname{Frob}_{p})$
; and– if
$\bar{r}$
is irreducible, then
$\unicode[STIX]{x1D706}=0$
; and– if
$\bar{r}$
is reducible, then we can write
$\bar{r}\cong \unicode[STIX]{x1D714}^{a+b}\otimes (\!\begin{smallmatrix}\overline{\unicode[STIX]{x1D712}}_{1} & \ast \\ 0 & \overline{\unicode[STIX]{x1D712}}_{2}\unicode[STIX]{x1D714}^{-b-1}\end{smallmatrix}\!)$
for unramified characters
$\overline{\unicode[STIX]{x1D712}}_{1}$
,
$\overline{\unicode[STIX]{x1D712}}_{2}$
and let
$\unicode[STIX]{x1D706}=\overline{\unicode[STIX]{x1D712}}_{1}(\operatorname{Frob}_{p})$
,
then the composition
maps$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:{\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })\rightarrow R_{p}(\unicode[STIX]{x1D70E})\rightarrow \mathbb{F}\end{eqnarray}$$
$T\mapsto \unicode[STIX]{x1D706}$
and
$S\mapsto \unicode[STIX]{x1D707}$
.(iv) Let
$\widehat{{\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })}$
be the completion of
${\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })$
with respect to the kernel of
$\unicode[STIX]{x1D6FC}$
. Then the map
${\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })\rightarrow R_{p}(\unicode[STIX]{x1D70E})$
induces an isomorphism of local
${\mathcal{O}}$
-algebras
$\widehat{{\mathcal{H}}(\unicode[STIX]{x1D70E}^{\circ })}\overset{\cong }{\rightarrow }R_{p}(\unicode[STIX]{x1D70E})$
. In coordinates, we have
$R_{p}(\unicode[STIX]{x1D70E})={\mathcal{O}}[[S-\widetilde{\unicode[STIX]{x1D707}},T-\widetilde{\unicode[STIX]{x1D706}}]]$
, where the tilde denotes the Teichmüller lift.(v) If we set
$\unicode[STIX]{x1D70B}:=(\operatorname{c - Ind}_{K}^{G}\overline{\unicode[STIX]{x1D70E}})\otimes _{{\mathcal{H}}(\overline{\unicode[STIX]{x1D70E}}),\unicode[STIX]{x1D6FC}}\mathbb{F}$
, then
$\unicode[STIX]{x1D70B}$
is an absolutely irreducible representation of
$G$
and is independent of the choice of Serre weight
$\overline{\unicode[STIX]{x1D70E}}$
of
$\bar{r}$
.
Proof. The claim that
$\bar{r}$
has at most two Serre weights is immediate from the proof of [Reference Buzzard, Diamond and JarvisBDJ10, Theorem 3.17], which explicitly describes the Serre weights of
$\bar{r}$
. Concretely, in the case at hand these weights are as follows (see also the discussion of [Reference EmertonEme11, § 3.5], which uses the same conventions as this paper). If
$\bar{r}$
is irreducible, then we may write
$$\begin{eqnarray}\bar{r}|_{I_{\mathbb{Q}_{p}}}\cong \unicode[STIX]{x1D714}^{m-1}\otimes \left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D714}_{2}^{n+1} & 0\\ 0 & \unicode[STIX]{x1D714}_{2}^{p(n+1)}\end{array}\right),\end{eqnarray}$$
where
$\unicode[STIX]{x1D714}_{2}$
is a fundamental character of niveau
$2$
, and
$0\leqslant m<p-1$
,
$0\leqslant n\leqslant p-2$
. Then the Serre weights of
$\bar{r}$
are
$\overline{\unicode[STIX]{x1D70E}}_{m,n}$
and
$\overline{\unicode[STIX]{x1D70E}}_{m+n,p-1-n}$
(with
$m+n$
taken modulo
$p-1$
). If
$\bar{r}$
is reducible, then we may write
$$\begin{eqnarray}\bar{r}|_{I_{\mathbb{Q}_{p}}}\cong \unicode[STIX]{x1D714}^{m+n}\otimes \left(\begin{array}{@{}cc@{}}1 & \ast \\ 0 & \unicode[STIX]{x1D714}^{-n-1}\end{array}\right),\end{eqnarray}$$
where
$0\leqslant m<p-1$
,
$0\leqslant n<p-1$
. Then (under Assumption 2.2), if
$n\neq 0$
, the unique Serre weight of
$\bar{r}$
is
$\overline{\unicode[STIX]{x1D70E}}_{m,n}$
, while, if
$n=0$
, then
$\overline{\unicode[STIX]{x1D70E}}_{m,0}$
and
$\overline{\unicode[STIX]{x1D70E}}_{m,p-1}$
are the two Serre weights of
$\bar{r}$
.
Part (2) follows from (1) by Lemma 2.14(2) and Corollary 2.11. We prove parts (1), (3) and (4) simultaneously. If
$\overline{\unicode[STIX]{x1D70E}}$
is not of the form
$\overline{\unicode[STIX]{x1D70E}}_{a,p-1}$
, the claims about
$R_{p}(\unicode[STIX]{x1D70E})$
are a standard consequence of (unipotent) Fontaine–Laffaille theory; for example, the irreducible case with
${\mathcal{O}}=\mathbb{Z}_{p}$
is [Reference Fontaine and MazurFM95, Theorem B2], and the reducible case follows in the same way. The key point is that the corresponding weakly admissible modules are either reducible, or are uniquely determined by the trace and determinant of
$\unicode[STIX]{x1D711}$
, by Lemma 2.6. Concretely, if
$\bar{r}$
is irreducible, then the crystalline lifts of
$\bar{r}$
of Hodge type
$\unicode[STIX]{x1D70E}_{a,b}$
correspond exactly to the weakly admissible modules
$D(-(a+b),1-a,t,d)$
, where
$v_{p}(t)>-a-b$
and
$\overline{p^{2a+b-1}d}=\unicode[STIX]{x1D707}$
. The claimed description of the deformation ring then follows.
Similarly, if
$\bar{r}$
is reducible, then it follows from Fontaine–Laffaille theory (and Assumption 2.2) that any crystalline lift of Hodge type
$\overline{\unicode[STIX]{x1D70E}}_{a,b}$
is necessarily reducible and indecomposable, and one finds that these crystalline lifts correspond precisely to those weakly admissible modules with
$D(-(a+b),1-a,t,d)$
, where
$v_{p}(t)=-a-b$
,
$\overline{p^{a+b}t}=\unicode[STIX]{x1D706}$
and
$\overline{p^{2a+b-1}d}=\unicode[STIX]{x1D707}$
.
This leaves only the case that
$\overline{\unicode[STIX]{x1D70E}}$
is of the form
$\overline{\unicode[STIX]{x1D70E}}_{a,p-1}$
. In this case the result is immediate from the main result of [Reference Berger, Li and ZhuBLZ04], which shows that the above description of the weakly admissible modules continues to hold.
Finally, (5) is immediate from the main results of [Reference Barthel and LivnéBL94, Reference BreuilBre03a], together with the explicit description of
$\overline{\unicode[STIX]{x1D70E}}$
and
$\unicode[STIX]{x1D706}$
, established above. More precisely, in the case that
$\bar{r}$
is irreducible, the absolute irreducibility of
$\unicode[STIX]{x1D70B}$
is [Reference BreuilBre03a, Theorem 1.1] and its independence of the choice of
$\overline{\unicode[STIX]{x1D70E}}$
is [Reference BreuilBre03a, Theorem 1.3]. If
$\bar{r}$
is reducible and has only a single Serre weight, then the absolute irreducibility of
$\unicode[STIX]{x1D70B}$
is [Reference Barthel and LivnéBL94, Theorem 33(2)]. In the remaining case that
$\bar{r}$
is reducible and has two Serre weights, then Assumption 2.2 together with the explicit description of
$\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}$
above implies that
$\unicode[STIX]{x1D706}^{2}\neq \unicode[STIX]{x1D707}$
, and the absolute irreducibility of
$\unicode[STIX]{x1D70B}$
is again [Reference Barthel and LivnéBL94, Theorem 33(2)]. The independence of
$\unicode[STIX]{x1D70B}$
of the choice of
$\overline{\unicode[STIX]{x1D70E}}$
is [Reference Barthel and LivnéBL94, Corollary 36(2)(b)].◻
Remark 2.16. It follows from the explicit description of
$\unicode[STIX]{x1D70B}$
that it is either a principal series representation or supersingular, and neither one dimensional nor an element of the special series. (This would no longer be the case if we allowed
$\bar{r}$
to be a twist of an extension of the trivial character by the mod
$p$
cyclotomic character, when in fact
$\unicode[STIX]{x1D70B}$
would be an extension of a one-dimensional representation and a special representation, which would also depend on the Serre weight if
$\bar{r}$
is peu ramifié.)
Remark 2.17. If
$\unicode[STIX]{x1D70B}$
has central character
$\unicode[STIX]{x1D713}$
, then
$\det \bar{r}=\unicode[STIX]{x1D713}\unicode[STIX]{x1D714}^{-1}$
.
3 Patched modules and arithmetic actions
We now introduce the notion of an arithmetic action of (a power series ring over)