2.1 Numbers

We fix a prime number $p$ and an algebraic closure $\overline {\mathbb {Q}}_p$ of $\mathbb {Q}_p$. We write $\mathbb {C}_p$ for the completion of $\overline {\mathbb {Q}}_p$. We denote by $\breve {\mathbb {Q}}_p\subset \mathbb {C}_p$ the completion of the maximal unramified extension of $\mathbb {Q}_p$ in $\overline {\mathbb {Q}}_p$, and by $\breve {\mathbb {Z}}_p \subset \breve {\mathbb {Q}}_p$ the completion of the ring of integers in the maximal unramified extension. We write ${\overline {\mathbb {F}}_p}$ for $\breve {\mathbb {Z}}_p/p$, an algebraically closed extension of $\mathbb {F}_p = \mathbb {Z}_p/p$. There is a canonical identification $W({\overline {\mathbb {F}}_p})=\breve {\mathbb {Z}}_p$, where $W(\bullet )$ denotes Witt vectors.

We write $\mathbb {A}$ for the adèles of $\mathbb {Q}$, $\mathbb {A}_f$ for the finite adèles, and $\mathbb {A}_f^{(p)}$ for the finite prime-to-$p$ adèles. We write $\widehat {\mathbb {Z}}$ for the profinite completion of $\mathbb {Z}$ and $\widehat {\mathbb {Z}}^{(p)}$ for the prime-to-$p$ profinite completion. We have canonical identifications

\[ \widehat{\mathbb{Z}}=\prod_{\ell \text{ prime} } \mathbb{Z}_\ell \quad \text{and} \quad \widehat{\mathbb{Z}}^{(p)}=\prod_{\ell \neq p \text{ prime }} \mathbb{Z}_\ell \]

and inclusions $\widehat {\mathbb {Z}} \subset \mathbb {A}_f$ and $\widehat {\mathbb {Z}}^{(p)} \subset \mathbb {A}_f^{(p)}$ inducing isomorphisms

\[ \widehat{\mathbb{Z}} \otimes \mathbb{Q} = \mathbb{A}_f \quad \text{and} \quad \widehat{\mathbb{Z}}^{(p)} \otimes \mathbb{Q} = \widehat{\mathbb{Z}}^{(p)} \otimes \mathbb{Z}_{(p)} = \mathbb{A}_f^{(p)}. \]

Here $\mathbb {Z}_{(p)}$ is interpreted via the notation $R_{\mathfrak {p}}$ for $R$ a ring and $\mathfrak {p}$ a prime ideal of $R$, which means the localization of $R$ by the multiplicative system $R\backslash \mathfrak {p}$.

2.2 Topological spaces, lisse sheaves, and torsors

Given a topological space $T$, we denote by $\underline {T}$ the topological constant sheaf with value $T$, that is, the functor on schemes

\[ \underline{T}(S) = \mathrm{Cont}(|S|, T) \]

where $|S|$ denotes the underlying topological space of $S$. When $T$ is a profinite set, it is represented by $\operatorname {SpecCont}(T, \mathbb {Z})$, where $\mathbb {Z}$ is equipped with the discrete topology (so that the continuous functions are just locally constant). It is, in particular, a sheaf for the pro-étale topology of [Reference Bhatt and ScholzeBS15].

We also adopt the framework of [Reference Bhatt and ScholzeBS15] as our formalism for lisse adelic sheaves.Footnote ³ Thus, by a lisse $\mathbb {A}_f$-sheaf on a scheme $S$, we mean a locally free of finite rank $\underline {\mathbb {A}_f}$ module on $S_\mathrm {pro\acute {e}t}$, and similarly for $\widehat {\mathbb {Z}}$, $\mathbb {A}_f^{(p)}$, $\widehat {\mathbb {Z}}^{(p)}$, $\mathbb {Q}_p,$ $\mathbb {Z}_p$, etc. A lisse $\widehat {\mathbb {Z}}$-sheaf is equivalent to a compatible system of locally free $\underline {\mathbb {Z}/n\mathbb {Z}}$ modules of finite rank (on either $S_\mathrm {\acute {e}t}$ or $S_\mathrm {pro\acute {e}t}$: a locally free $\underline {\mathbb {Z}/n\mathbb {Z}}$ module on $S_\mathrm {pro\acute {e}t}$ is automatically classical because $\mathbb {Z}/n\mathbb {Z}$ is discrete), and similarly for $\widehat {\mathbb {Z}}^{(p)}$ and $\mathbb {Z}_p$.

For $K$ a topological group, a (right) $K$-torsor on $S_\mathrm {pro\acute {e}t}$ is a sheaf $\mathcal {K}$ equipped with an action $\mathcal {K} \times \underline {K} \rightarrow \mathcal {K}$ such that locally on $S_\mathrm {pro\acute {e}t}$, $\mathcal {K} \cong \underline {K}$ with the action by right multiplication. In particular, if $A=\mathbb {A}_f, \widehat {\mathbb {Z}}, \mathbb {A}_f^{(p)}, \widehat {\mathbb {Z}}^{(p)}, \mathbb {Z}_p,\text {or } \mathbb {Q}_p,$ and $V$ is a lisse $A$-sheaf of rank $n$, then

\[ \underline{\mathrm{Isom}}(\underline{A}^n, V) := T \mapsto \mathrm{Isom}(\underline{A}^n_T, V_T) \]

is a $\mathrm {GL}_n(A)$-torsor (as in these cases $\underline {\mathrm {Isom}}(\underline {A}^n, \underline {A}^n)= \underline { \mathrm {GL}_n(A) })$.

The following lemma will be used as a technical tool for moving between infinite-level and finite-level moduli problems for elliptic curves. When $G=\mathrm {GL}_m(\mathbb {Z}_\ell )$ and $H=\{e\}$, it amounts to the statement that a rank $m$ lisse $\mathbb {Z}_\ell$-sheaf is the same as a compatible family of rank $m$ lisse $\mathbb {Z}/\ell ^n\mathbb {Z}$-sheafs, which will surprise nobody.

Lemma 2.2.1 Let $G$ be a profinite group, $H\leq G$ a closed subgroup, and $\mathcal {G}$ a $G$-torsor on $\operatorname {Spec}\!R_\mathrm {pro\acute {e}t}$. The map

(2.2.1)\begin{equation} \mathcal{H} \mapsto (\mathcal{H} \cdot \underline{U})_{H \leq U \leq G,\; U \text{compact open} } \end{equation}

is a bijection between the set of $H$-torsors in $\mathcal {G}$ and compatible systems of $U$-torsors in $G$ for $H \leq U \leq G,$ $U$ a compact open subgroup.

Proof. It suffices to consider a cofinal system of $U$; thus we take a neighborhood basis of the identity in $G$ consisting of open normal subgroups $G_\epsilon$, $\epsilon \in I$, and consider only $U$ of the form $H_\epsilon := H \cdot G_\epsilon$. Note that $\bigcap _{\epsilon \in I} H_\epsilon = H$.

The key point is to show that if $(\mathcal {H}_\epsilon \subset \mathcal {G})_\epsilon$ is a compatible system of $H_\epsilon$-torsors, then $\bigcap _{\epsilon \in I} \mathcal {H}_\epsilon$ admits a section on a pro-étale cover. Indeed, then because $\bigcap _{\epsilon \in I}\underline {H_\epsilon } = \underline {H}$, $\bigcap _{\epsilon \in I} \mathcal {H}_\epsilon$ will automatically be an $H$-torsor, and it is straightforward to check this is a two-sided inverse to (2.2.1).

For this key point, by passing to a pro-étale cover we may assume $\mathcal {G}$ is trivial, i.e. we can take $\mathcal {G}=\underline {G}$. We now choose a compatible family of splittings of $G \rightarrow G/H_\epsilon$ (this is possible because for each $\epsilon$ the set of splittings is a finite set and the transition maps are surjective), thus we obtain a compatible family of homeomorphisms, each equivariant for the right multiplication actions of $H_\epsilon$,

\[ G = G/H_\epsilon \times H_\epsilon \]

and, passing to the limit over $\epsilon$, a homeomorphism

\[ G = G/H \times H \]

equivariant for the right multiplication action of $H$.

From this it follows that the $\underline {H_\epsilon }$-torsors in $\mathcal {G}$ are identified with $\underline {G/H_\epsilon }(\operatorname {Spec}\!R)$, the $\underline {H}$-torsors in $\mathcal {G}$ are identified with $\underline {G/H}(\operatorname {Spec}\!R)$, and the map $\mathcal {H} \mapsto \mathcal {H} \cdot \underline {H_\epsilon }$ is induced by the canonical projection $G/H \rightarrow G/H_\epsilon$. The result then follows as $G/H = \lim _{\epsilon \in I} G/H_\epsilon$.

2.3 Elliptic curves and quasi-isogenies

For $R$ a ring, we will consider the category $\operatorname {Ell}(R)$ of elliptic curves over $R$. It is $\mathbb {Z}$-linear. The isogeny category

\[ \operatorname{Ell}(R) \otimes \mathbb{Q} \]

has the same objects but homomorphisms are tensored with $\mathbb {Q}$. For $E$ an elliptic curve, we sometimes write $E \otimes \mathbb {Q}$ for the corresponding element of $\operatorname {Ell}(R) \otimes \mathbb {Q}$, so

\[ \mathrm{Hom} (E_1 \otimes \mathbb{Q}, E_2 \otimes \mathbb{Q}) = \mathrm{Hom}(E_1, E_2) \otimes \mathbb{Q}. \]

An isogeny from $E_1$ to $E_2$ is a morphism $f: E_1 \rightarrow E_2$ such that $f \otimes \mathbb {Q}: E_1 \otimes \mathbb {Q} \rightarrow E_2 \otimes \mathbb {Q}$ is invertible. A quasi-isogeny from $E_1$ to $E_2$ is an invertible morphism $f:E_1 \otimes \mathbb {Q} \rightarrow E_2 \otimes \mathbb {Q}$; we often write instead, e.g., ‘$f:E_1 \rightarrow E_2$ is a quasi-isogeny’.

Similarly, we consider the prime-to-$p$ isogeny category $\operatorname {Ell}(R) \otimes \mathbb {Z}_{(p)}$ by replacing $\mathbb {Q}$ everywhere above with $\mathbb {Z}_{(p)}$. A prime-to-$p$ isogeny from $E_1$ to $E_2$ is a morphism $f: E_1 \rightarrow E_2$ such that $f \otimes \mathbb {Z}_{(p)}: E_1 \otimes \mathbb {Z}_{(p)} \rightarrow E_2 \otimes \mathbb {Z}_{p}$ is invertible. A prime-to-$p$ quasi-isogeny from $E_1$ to $E_2$ is an invertible morphism $f:E_1 \otimes \mathbb {Z}_{(p)} \rightarrow E_2 \otimes \mathbb {Z}_{(p)}$; we often write instead, e.g., ‘$f:E_1\rightarrow E_2$ is a prime-to-$p$ quasi-isogeny’.

2.3.2 Tate modules

If $R/\mathbb {Q}$ (i.e. $R$ is of characteristic zero) and $E/R$ is an elliptic curve, we consider the $p$-adic and adelic integral and rational Tate modules

\[ T_p(E):= \lim_n E[p^n],\quad V_p(E):=T_p(E)[1/p],\quad T_{\widehat{\mathbb{Z}}}(E) := \lim_n E[n], \quad V_{\mathbb{A}_f}(E):=T_{\widehat{\mathbb{Z}}}(E) \otimes \mathbb{Q}. \]

These are lisse rank-two sheaves on $\operatorname {Spec}\!R$ over $\mathbb {Z}_p, \mathbb {Q}_p, \widehat {\mathbb {Z}},$ and $\mathbb {A}_f$, respectively. All are functors on $\operatorname {Ell}(R)$, and $V_p$ and $V_{\mathbb {A}_f}$ factor through $\operatorname {Ell}(R) \otimes \mathbb {Q}$.

If $R/\mathbb {Z}_{(p)}$ (i.e. all primes $\ell \neq p$ are invertible in $R$) and $E/\operatorname {Spec}\!R$ is an elliptic curve, then we may still form the prime-to-$p$ integral and adelic Tate modules

\[ T_{\widehat{\mathbb{Z}}^{(p)}}(E) := \lim_n E[n] \quad\mbox{and}\quad V_{\mathbb{A}_f^{(p)}}(E):=T_{\widehat{\mathbb{Z}}^{(p)}}(E) \otimes \mathbb{Q} = T_{\widehat{\mathbb{Z}}^{(p)}}(E) \otimes \mathbb{Z}_{(p)}. \]

These are lisse rank-two sheaves on $\operatorname {Spec}\!R$ over $\widehat {\mathbb {Z}}^{(p)}$ and $\mathbb {A}_f^{(p)}$, respectively. Both are functors on $\operatorname {Ell}(R)$, and $V_{\mathbb {A}_f^{(p)}}$ factors through $\operatorname {Ell}(R) \otimes \mathbb {Z}_{(p)}$.

2.3.3 Relative differentials

For $\pi :E\rightarrow \operatorname {Spec}\!R$ an elliptic curve, $\omega _{E/R}:=\pi _* \Omega _{E/R}$ is a line bundle on $S$. Restriction induces canonical isomorphisms

\[ \omega_{E/R} = 1_E^*\Omega_{E/R} = (\operatorname{Lie} E/R)^*, \]

where, here, $1_E: \operatorname {Spec}\!R \rightarrow E$ is the identity section.

The assignment $E/R \rightarrow \omega _{E/R}$ is a functor from $\operatorname {Ell}(R)$ to line bundles on $\operatorname {Spec}\!R$. If $R/\mathbb {Q}$, then it factors through $\operatorname {Ell}(R) \otimes \mathbb {Q}$, and if $R/\mathbb {Z}_{(p)}$, it factors through ${\operatorname {Ell}(R) \otimes \mathbb {Z}_{(p)}}$; indeed, for $n \in \mathbb {Z}$, the multiplication map by $n$ map $[n]:E \rightarrow E$ induces ring multiplication by $n$ on $\omega _{E/R}$, thus is invertible if $n$ is invertible in $R$.

2.3.4 $p$-divisible groups

For $R$ a ring, a $p$-divisible group $G$ of height $h \in \underline {\mathbb {N}}(\operatorname {Spec}\!R)$ is, following Tate [Reference TateTat67, 2.1] and Messing [Reference MessingMes72, I.2], an inductive system

\[ (G_i, \iota_i),\quad i \geq 0 \]

of finite locally free commutative group schemes $G_i$ of degree $p^{ih}$ over $\operatorname {Spec}\!R$, equipped with closed immersions $\iota _i: G_i \rightarrow G_{i+1}$ identifying $G_i$ with the kernel of multiplication by $p^i$ on $G_{i+1}$.

We write $p\text {-}\mathrm {div}(R)$ for the $\mathbb {Z}_p$-linear category of $p$-divisible groups over $R$. There is a natural functor

\[ \operatorname{Ell}(R) \rightarrow p\text{-}\mathrm{div}(R): E/R \mapsto E[p^\infty]:=(E[p^i])_i. \]

This functor factors through $\operatorname {Ell}(R) \otimes \mathbb {Z}_{(p)}$. We also form the isogeny category $p\text {-}\mathrm {div}(R) \otimes \mathbb {Q}_p$ and define isogenies and quasi-isogenies in the obvious way. The functor $E \mapsto E[p^\infty ] \otimes \mathbb {Q}_p$ then factors through $\operatorname {Ell}(R) \otimes \mathbb {Q}$.

If $R$ is a $p$-adically complete ring, then we write $\mathrm {Nilp}_R$ for the category of $R$-algebras where $p$ is nilpotent and we view a $p$-divisible group $G=(G_i)$ over $R$ as the functor on $\mathrm {Nilp}_R$

\[ G(A) = \mathrm{colim}_i\, G_i(A). \]

In this case, because $E[p^\infty ]_{R/p^n}$ and $E_{R/p^n}$ have the same tangent space for any $n$ and $R$ is $p$-adically complete, the functor $E \mapsto \omega _{E/S}$ factors through $E \mapsto E[p^\infty ]$.

2.4 Completing algebra actions

In this section, we develop the basic definitions for completing algebra actions and some tools for comparing completions. This material is used in § 5.7, where it is essential for the final deduction of Theorem A from the key geometric input, Corollary 5.6.3. Some results are also used in § 4.5 when we explain how Theorem 1.1.1 can be deduced from Theorem A.

The statements we give here are likely well known to experts and the proofs are, for the most part, elementary exercises in analysis. Nonetheless we include a full treatment because we are not aware of another suitable source in the literature.

We begin with some basic definitions in non-archimedean functional analysis. In the following, $L$ is any complete non-archimedean field.

Note that because in Definition 2.4.1(ii) we assumed $\{e_i\}_{i \in I}$ was bounded, all sums of the form (2.4.1) converge, and then the open mapping theorem implies that the sup norm $|v|=\sup _{i \in I} |v_i|_L$ is a Banach norm. For $L$ discretely valued (or, more generally, spherically complete), every $L$-Banach space is orthonormalizable; cf. [Reference SerreSer62, Corollaire of Proposition 1 and Remarques after Proposition 2].

The topology of pointwise convergence is uniquely determined by the property that a net $(T_j)_{j \in J}$ in $B(V,W)$ converges to $T\in B(V,W)$ if and only if $T_j(v) \rightarrow T(v)$ for all $v \in V$. It is through this characterization that we access it.

The following lemma is elementary but extremely useful.

Lemma 2.4.4 Suppose $V$ and $W$ are $L$-Banach spaces, $S \subset V$ is such that the set $L[S]$ of finite linear combinations of elements of $S$ is dense in $V$, $(T_{j})_{j \in J}$ is a bounded net of operators in $B(V,W)$, and $T \in B(V,W)$. Then $T_j \rightarrow T$ in the in the topology of pointwise convergence if and only if

(2.4.2)\begin{equation} \lim_{j \in J} T_j(v)=T(v) \quad \text{for all } v \in S. \end{equation}

Proof. As previously, we have $T_j \rightarrow T$ in the topology of pointwise converge if and only if, for every $v \in V$, $\lim _{j \in J} T_j(v)=T(v)$. Thus, one direction is immediate. For the other, suppose that (2.4.2) holds and fix Banach norms on $V$ and $W$. By the boundedness hypothesis, we can then choose a common bound $C\geq 1$ for the operator norms of all $T_j,\ j \in J$, and $T$.

Let $v \in V$. By the density hypothesis, for any $\epsilon > 0$ we can find

\[ v'=\ell_1 v_1 + \cdots + \ell_k v_k, v_i \in S \]

such that $|v-v'| \leq \epsilon$. By (2.4.2), for each $v_i$, there is a $j_i \in J$ such that, for $j\geq j_i$,

\[ |T(\ell_i v_i)-T_j(\ell_i v_i)|=|\ell_i||T(v_i)-T_j(v_i)| \leq \epsilon. \]

Thus, taking $j' \geq j_1, \ldots, j_k$ (the fundamental property of the directed set indexing a net is that there is always an upper bound for any finite collection of elements), we obtain that for $j \geq j'$,

\begin{align*} |T(v)-T_j(v)| &= | (T(v') - T_j(v')) + T(v-v') - T_j(v-v')|\\ &= \biggl|\sum_{i=1}^k(T(\ell_i v_i)-T_j(\ell_i v_i)) + T(v-v') - T_j(v-v')\biggr|\\ &\leq \max(\epsilon, |T(v-v')|, |T_j(v-v')|) \leq C\epsilon. \end{align*}

We conclude that $\lim _{j\in J} T_j(v) = T(v)$, as desired.

Definition 2.4.5 If $A$ is a ring, $L$ is a non-archimedean field, and $(W_i)_{i \in I}$ is family of $L$-Banach spaces equipped with actions of $A$ by operators in $B(W_i,W_i)$, the completion Footnote ⁵ of $A$ acting on $(W_i)_{i \in I}$ is the closure $\widehat {A}_{(W_i)_{i \in I}}$ of the image of $A$ in

\[ \prod_{i \in I} B(W_i, W_i), \]

where $B(W_i, W_i)$ is equipped with the topology of pointwise convergence and the product is equipped with the product topology. Concretely, $(T_i)_{i \in I} \in \widehat {A}_{(W_i)_{i \in I}}$ if and only if there exists a net $(a_j)_{j \in J}$ in $A$ whose image converges to $(T_i)_{i \in I}$. The latter is equivalent to asking that, for any $i \in I$ and any $w_i \in W_i$,

\[ \lim_{j \in J} a_j \cdot w_i = T_i(w_i). \]

Proof. It is always a closed subgroup, so it remains just to see that under the boundedness hypothesis it is closed under composition.

Suppose given $(T_i)_{i \in I}$ and $(S_i)_{i \in I}$ in $\widehat {A}_{(W_i)_{i \in I}}$, and choose nets $(a_{j_T})_{j_T \in J_T}$ and $(b_{j_S})_{j_{S} \in J_S}$ whose images in $\prod _{i \in I} \mathrm {End}(W_i)$ converge to $(T_i)_{i \in I}$ and $(S_i)_{i \in I}$, respectively. Then we claim that the image of

\[ (a_{j_T}b_{j_S})_{(j_T,j_S)\in J_T \times J_S}\]

converges to $(T_i \circ S_i)_{i \in I}$. It suffices to show that for any $i \in I$ and $w_i \in W_i$,

\[ \lim_{(j_T,j_S)\in J_T \times J_S} = a_{j_T}b_{j_S} \cdot w_i = T_i(S_i(w_i)). \]

We suppress the $i$ now and write $W_i=W$, $w_i=w$, $T_i=T$, and $S_i=S$.

To see the convergence, fix a Banach norm on $W$ and, by boundedness of the action, a $C\geq 1$ such that $|a \cdot v|\leq C|v|$ for all $a \in A$ and $v \in W$. Then, for any $\epsilon > 0$, we may choose $j_{T,0} \in j_T$ and $j_{S,0} \in J_S$ such that:

1. (i) $|a_{j_T} \cdot S(w) - T(S(w))| \leq \epsilon$ for all $j_T \geq j_{T,0}$; and

2. (ii) $|b_{j_S} \cdot w -S(w)| \leq \epsilon$ for all $j_S \geq j_{S,0}$.

Then, for $(j_S,j_T)\geq (j_{S,0}, j_{T,0})$,

\begin{align*} |a_{j_T}b_{j_S} \cdot w - T(S(w))| &= \big|a_{j_T}\cdot(b_{j_S} \cdot w - S(w)) + (a_{j_T}\cdot S(W) - T(S(w))) \big|\\ & \leq \max(|a_{j_T}\cdot(b_{j_S} \cdot w - S(w))|,\, |a_{j_T}\cdot S(W) - T(S(w))| ) \\ & \leq \max( C\epsilon, \epsilon ) \\ & \leq C\epsilon \end{align*}

and we conclude.

The following lemma allows for comparison with other definitions in the literature, in particular the definition given in [Reference EmertonEme14, 2.1.4].

Lemma 2.4.7 Using notation as before, suppose $L$ is discretely valued and write $\mathcal {O}_L$ for the ring of integers and $\mathfrak {p}$ for its maximal ideal. If each $W_i$ is finite dimensional and, for each $i$, $A$ preserves an $\mathcal {O}_L$-lattice $W_i^\circ \subset W_i$, then the action is bounded and $\widehat {A}_{(W_i)_{i \in I}}$ is naturally identified with the closure of the image of $A$ in

\[ \prod_{i \in I, n>0} \mathrm{End}_{\mathcal{O}_K} (W_i^\circ / \mathfrak{p}^n W_i^\circ) \]

equipped with the product topology (each term is equipped with the discrete topology).

Proof. Boundedness is clear. For the rest, first note the image of $A$ in $\prod _{i \in I} B(W_i, W_i)$ factors through $\prod _{i \in I} \mathrm {End}_{\mathcal {O}_K}(W_i^\circ )$, where we identify $\mathrm {End}_{\mathcal {O}_K}(W_i^\circ )$ with the subset of $B(W_i, W_i)$ preserving $W_i^\circ$. This subset is closed, so we can form $\widehat {A}_{V}$ by taking the closure of the image of $A$ in $\prod _{i \in I} \mathrm {End}_{\mathcal {O}_K}(W_i^\circ )$. Then, for each $i$ we have

\[ \mathrm{End}_{\mathcal{O}_K}(W_i^\circ)= \lim_n \mathrm{End}_{\mathcal{O}_K} (W_i^\circ / \mathfrak{p}^n W_i^\circ), \]

where each term on the right is equipped with the discrete topology. Thus,

\[ \prod_{i \in I} \mathrm{End}_{\mathcal{O}_K}(W_i^\circ) \subset \prod_{i \in I, n>0} \mathrm{End}_{\mathcal{O}_K} (W_i^\circ / \pi^n) \]

is closed so we may compute $\widehat {A}_V$ by taking the closure in the space on the right.

The following lemma says that completion is insensitive to base extension. This is useful as our comparisons of Hecke modules take place over very large extensions of $\mathbb {Q}_p$, whereas one is typically interested in Hecke algebras over $\mathbb {Z}_p$.

Lemma 2.4.8 Let $L \subset L'$ be an extension of complete non-archimedean fields, and let $A$ be a ring. Suppose $(W_i)$ is a family of orthonormalizable $L$-Banach spaces equipped with bounded actions of $A$. Then the identity map $A \rightarrow A$ extends uniquely to a topological isomorphism

\[ \widehat {A}_{(W_i)_{i\in I}} = \widehat {A}_{(W_i \widehat{\otimes}_L L')_{i \in I}}. \]

The following is our main technical tool for comparing completed Hecke algebras.

The following lemma combines some of the results given previously, and is used in § 4.5 to deduce Theorem 1.1.1 from Theorem A.

Lemma 2.4.11 Suppose $L$ is discretely valued and write $\mathcal {O}_L$ for the ring of integers and $\mathfrak {p}$ for the maximal ideal. Suppose $V$ is an $L$-Banach space and $A$ acts on $V$ preserving a bounded open $\mathcal {O}_K$-lattice $V^\circ$, and $(W_i)_{i \in I}$ is a filtered system of finite-dimensional $A$-invariant subspaces such that $\bigcup _{i \in I} W_i$ is dense in $V$. Then, writing

\[ W_i^\circ = W_i \cap V^\circ, \quad \overline{W}_{i,n}=W_i ^\circ / \mathfrak{p}^n W_i^\circ = W_i^\circ/ W_i \cap \mathfrak{p}^n V^\circ, \]

and $A_{i,n}$ for the image of $A$ in $\mathrm {End}_{\mathcal {O}_K}( \overline {W}_{i,n})$, we have

\[ \widehat{A}_V = \widehat{A}_{(W_i)_{i \in I}} = \lim_{(i,n) \in I \times \mathbb{N}} A_{i,n}, \]

where each term in the limit is equipped with the discrete topology.

3.1 Modular curves

Definition 3.1.1 (The level $K$ elliptic moduli functor)

Let $K \subset \mathrm {GL}_2(\mathbb {A}_f)$ be a closed subgroup. Let $Y_K$ be the functor on $\mathbb {Q}$-algebras

\[ Y_K: R \mapsto \{ (E, \mathcal{K}) \} /\sim \]

sending $R/\mathbb {Q}$ to the set of equivalence classes of pairs $(E, \mathcal {K})$ where:

1. (i) $E/R$ is an elliptic curve;

2. (ii) $\mathcal {K} \subset \underline {\mathrm {Isom}}( (\underline {\mathbb {A}_f})^2, V_{\mathbb {A}_f }(E))$ is a $K$-torsor;

3. (iii) the relation $\sim$ is defined by $(E,\mathcal {K}) \sim (E', \mathcal {K}')$ if there is a quasi-isogeny $q: E \rightarrow E'$ such that $q(\mathcal {K}) = \mathcal {K}'$.

The topological constant sheaf on the normalizer of $K$, $\underline {N_{\mathrm {GL}_2(\mathbb {A}_f)}(K)}$, acts on $Y_K$, and for $K_1 \leq K_2$ we have the obvious map

\[ Y_{K_1} \rightarrow Y_{K_2}, \quad (E, \mathcal{K}_1) \mapsto (E, \mathcal{K}_1 \cdot \underline{K_2} ). \]

Example 3.1.2 (Infinite level)

Take $K=\{e\}$. Then the $K$-torsor $\mathcal {K}$ appearing in the moduli problem $Y_{K}$ is just a section of

\[ \underline{\mathrm{Isom}}( \underline{\mathbb{A}_f}^2, V_{\mathbb{A}_f}(E)), \]

i.e. an isomorphism $\varphi _{\mathbb {A}_f}:\underline {\mathbb {A}_f}^2 \xrightarrow {\sim } V_{\mathbb {A}_f}(E)$, and the condition in the equivalence relation becomes $q \circ \varphi _{\mathbb {A}_f} = \varphi '_{\mathbb {A}_f}$. The group action is by all of $\underline {\mathrm {GL}_2(\mathbb {A}_f)}$, and in this notation it acts by composition with $\varphi _{\mathbb {A}_f}$. When $K=\{e\}$, we typically omit it from the notation and write simply $Y=Y_{\{e\}}$.

Removing any level structure at $p$, we obtain a variant over $\mathbb {Z}_{(p)}$.

Definition 3.1.3 (The integral level $K^p$ elliptic moduli functor)

Let $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$ be a closed subgroup. Let $\mathfrak {Y}_{K^p}$ be the functor on $\mathbb {Z}_{(p)}$-algebras

\[ \mathfrak{Y}_{K^p}: R \mapsto \{ (E, \mathcal{K}^p) \} /\sim \]

sending $R/\mathbb {Z}_{(p)}$ to the set of equivalence classes of pairs $(E, \mathcal {K}^p)$ where:

1. (i) $E/R$ is an elliptic curve;

2. (ii) $\mathcal {K}^p \subset \underline {\mathrm {Isom}}( (\underline {\mathbb {A}_f^{(p)}})^2, V_{\mathbb {A}_f^{(p)}}(E))$ is a $K^p$-torsor;

3. (iii) the relation $\sim$ is defined by $(E, \mathcal {K}^p) \sim (E', {\mathcal {K}^p}')$ if there is a prime-to-$p$ quasi-isogeny $q: E \rightarrow E'$ such that $q(\mathcal {K}^p) = {\mathcal {K}^p}'$.

The topological constant sheaf on the normalizer of $K$, $\underline {N_{\mathrm {GL}_2(\mathbb {A}_f^{(p)})}(K^p)}$, acts on $\mathfrak {Y}_{K^p}$, and for $K^p_1 \leq K^p_2$ we have the obvious map

\[ \mathfrak{Y}_{K^p_1} \rightarrow \mathfrak{Y}_{K^p_2}, \quad (E, \mathcal{K}_1^p) \mapsto (E, \mathcal{K}_1^p \cdot \underline{K^p_2}). \]

Arguing as in [Reference DeligneDel71, Corollaire 3.5], we find the following.

Lemma 3.1.5 Let $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$ be a closed subgroup. The assignment

\[ (E, \mathcal{K}^p) \rightarrow (E, \underline{\mathrm{Isom}}(\mathbb{Z}_p^2, T_p E) \times \mathcal{K}^p) \]

induces an isomorphism

\[ \mathfrak{Y}_{K^p, \mathbb{Q}} \xrightarrow{\sim} Y_{\mathrm{GL}_2(\mathbb{Z}_p)K^p}. \]

Note that if $K_2 \leq K_1 \leq \mathrm {GL}_2(\mathbb {A}_f)$ (respectively, $K_2^p \leq K_1^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$), are closed subgroups and $K_1$ (respectively, $K_1^p$) is sufficiently small, then so is $K_2$ (respectively, $K_2^p$). Moreover, the property of being a sufficiently small closed subgroup of $\mathrm {GL}_2(\mathbb {A}_f)$ (respectively, $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$) is preserved under conjugation by $\mathrm {GL}_2(\mathbb {A}_f)$ (respectively, $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$). Because any lattice $\mathcal {L}$ is in the $\mathrm {GL}_2(\mathbb {A}_f)$-orbit of $\widehat {\mathbb {Z}}^2$ (respectively, $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-orbit of $(\widehat {\mathbb {Z}}^{(p)})^2$), being sufficiently small is equivalent to being contained in a conjugate of the standard principal congruence subgroup of level $n \geq 3$ (respectively, $(n,p)=1$).

The main representability results are as follows.

Proposition 3.1.8 If $K \leq \mathrm {GL}_2(\mathbb {A}_f)$ (respectively, $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$) is a sufficiently small closed subgroup, then $Y_K$ (respectively, $\mathfrak {Y}_{K^p}$) is represented by an affine scheme over $\operatorname {Spec}\!\mathbb {Q}$ (respectively, $\operatorname {Spec}\!\mathbb {Z}_{(p)}$), and the natural map

(3.1.1)\begin{equation} Y_K \rightarrow \lim_{\substack{K' \text{ compact open}\\ K \leq K' \leq \mathrm{GL}_2(\mathbb{A}_f)}} Y_{K'} \quad \text{ (respectively, } \mathfrak{Y}_{K^p} \rightarrow \lim_{\substack{{K^p}' \text{ compact open}\\ {K^p} \leq {K^p}' \leq \mathrm{GL}_2(\mathbb{A}_f^{(p)})}} \mathfrak{Y}_{{K^p}'} \text{)} \end{equation}

is a $\underline {N_{\mathrm {GL}_2(\mathbb {A}_f)}(K)}$-equivariant (respectively, $\underline {N_{\mathrm {GL}_2(\mathbb {A}_f^{(p)})}(K^p)}$-equivariant) isomorphism, where the action on the right-hand side is induced by the action on the tower that permutes the terms by conjugation (i.e. right multiplication by $h$ sends $Y_{K'}$ to $Y_{h^{-1}K'h}$).

If $K$ (respectively, $K^p$) is furthermore compact open, then $Y_K$ (respectively, $\mathfrak {Y}_{K^p}$) is a smooth affine curve. Moreover, for $K_1 \leq K_2$ (respectively, $K_1^p \leq K_2^p$) sufficiently small closed subgroups the natural map $Y_{K_1} \rightarrow Y_{K_2}$ (respectively, $\mathfrak {Y}_{K_1^p} \rightarrow \mathfrak {Y}_{K_2^p}$) is profinite étale, and, if $K_1 \trianglelefteq K_2$ (respectively, $K_1^p \trianglelefteq K_2^p$) it is Galois with group $K_1/K_2$ (respectively, $K_1^p/K_2^p$).

We refer to the boundary $X_K\backslash Y_K$ (respectively, $\mathfrak {X}_{K^p} \backslash \mathfrak {Y}_{K^{p}}$) with its reduced subscheme structure as the cusps. The cusps can also be described as filling in the punctures corresponding to level structure on the Tate curve as in [Reference Katz and MazurKM85, 8.11].

3.2 Modular forms

For any sufficiently small closed $K \leq \mathrm {GL}_2(\mathbb {A}_f)$ (respectively, $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$), we have a universal elliptic curve $E_K/Y_K$ (respectively, $\mathfrak {E}_{K^p} / \mathfrak {Y}_{K^p}$), determined up to unique isogeny (respectively, prime-to-$p$ isogeny). We write simply $\omega$ for the line bundle $\omega _{E/K} / Y_K$ (respectively, $\omega _{\mathfrak {E}_{K^p}/\mathfrak {Y}_{K^p}}/\mathfrak {Y}_{K^p}$): it is determined up to unique isomorphism compatibly with all pullbacks, base change, and group actions discussed so far, so that this notation will cause no confusion.

For every sufficient small compact open $K$ (respectively, $K^p$), we extend $\omega$ to $X_K$ (respectively, to $\mathfrak {X}_{K^p}$) in the standard way by allowing sections with holomorphic $q$-expansions at each cusp. Direct computation shows this is compatible with all pullbacks, base change, and group actions discussed so far, so that we can extend this definition to any sufficiently small closed $K$ (respectively, $K^p$) and again no confusion will be caused by referring to the extended line bundle also as $\omega$.

We consider the smooth $\mathrm {GL}_2(\mathbb {A}_f)$-representation of modular forms,

\[ M_{k,\mathbb{Q}} = H^0(X, \omega^k). \]

For $K$ any sufficiently small closed subgroup, pullback from level $K$ identifies

\[ M_{k,\mathbb{Q}}^K = H^0(X_K, \omega^k). \]

Applied to $K$ sufficiently small compact open, we deduce that $M_{k,\mathbb {Q}}$ is an admissible representation of $\mathrm {GL}_2(\mathbb {A}_f)$. Applied to $K=\mathrm {GL}_2(\mathbb {Z}_p)$, we obtain

\[ M_{k,\mathbb{Q}}^{\mathrm{GL}_2(\mathbb{Z}_p)} = H^0(X_{\mathrm{GL}_2(\mathbb{Z}_p)}, \omega^k)=H^0(\mathfrak{X}_{\mathbb{Q}}, \omega^k)=H^0(\mathfrak{X}, \omega^k)\otimes_{\mathbb{Z}_{(p)}} \mathbb{Q}. \]

In particular, if we write

\[ M_{k,\mathbb{F}_p} = H^0(\mathfrak{X}_{\mathbb{F}_p}, \omega^k), \]

an admissible $\mathbb {F}_p$-representation of $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$ by the same argument as previously, then $H^0(\mathfrak {X}, \omega ^k)$ is a natural $\mathbb {Z}_{(p)}$-lattice in $M_{k,\mathbb {Q}}^{\mathrm {GL}_2(\mathbb {Z}_p)}$ equipped with a $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant reduction map to $M_{k,\mathbb {F}_p}$.

3.2.2 The Hasse invariant

We now recall how, to any elliptic curve $E/R$ for $R/\mathbb {F}_p$, one can attach a canonical section $\mathrm {Ha}(E/R) \in \omega ^{p-1}_{E/R}$, the Hasse invariant. We follow one of the approaches described in [Reference Katz and MazurKM85, 12.3].

As the section $\mathrm {Ha}(E/R)$ can be constructed Zariski locally, it suffices to assigns to any pair $(E/R, \alpha )$ where $R$ is an $\mathbb {F}_p$-algebra, $E/ R$ is an elliptic curve and $\alpha \in \omega _{E/R}$ is a non-vanishing invariant differential, an element $\mathrm {Ha}(E/ R,\alpha )$ of $R$ such that, for $a \in A^\times$,

\[ \mathrm{Ha}(E/ R, a \alpha)= a^{-(p-1)}\mathrm{Ha}(E/ R, \alpha) \]

and whose formation is functorial in base change and isomorphism. In this case, to give our rule we first take the invariant derivation $\partial _{\alpha }$ that is dual to $\alpha$, then form

\[ \partial_{\alpha}^p := \underbrace {\partial_{\alpha} \circ \cdots \circ \partial_{\alpha} }_{p\text{ times}}, \]

which is also an invariant derivation and thus a multiple of $\partial$. Then the equation

\[ \partial^p_\alpha = \mathrm{Ha}(E/ R, \alpha) \partial_\alpha, \]

defines $\mathrm {Ha}(E/ R, \alpha )$, and it is straightforward to check this satisfies the desired transformation rule if we scale $\alpha$ and is functorial in base change and isomorphism.

In fact, the construction is also functorial in prime-to-$p$ quasi-isogenies: it suffices to observe that a prime-to-$p$ quasi-isogeny induces an isomorphism of $p$-divisible groups and, in particular, of formal groups, and that the action of an invariant derivation is completely determined by its action on the formal group. This observation also has the important consequence that the resulting section $\mathrm {Ha}(E/R)$ can be constructed entirely in terms of $E[p^\infty ]$.

Applying this construction to the universal elliptic curve over $\mathfrak {Y}_{\mathbb {F}_p}$, we obtain a $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-invariant section of $\omega ^{p-1}$. A direct computation on the Tate curve (see [Reference Katz and MazurKM85, Theorem 12.4.2]) shows that its $q$-expansions are constant equal to $1$ at every cusp, thus it extends to

\[ \mathrm{Ha} \in M_{p-1, \mathbb{F}_p}^{\mathrm{GL}_2(\mathbb{A}_f^{(p)})}. \]

3.3 Supersingular and ordinary loci

Let

\[ b_{\mathrm{ss}}= \begin{pmatrix} 0 & p \\ 1 & 0 \end{pmatrix} \in M_2(\mathbb{Z}_p) \quad \text{and}\quad b_\mathrm{ord}=\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix} \in M_2(\mathbb{Z}_p). \]

Let $\mathbb {X}_{\mathrm {ss}} / \mathbb {F}_p$ (respectively, $\mathbb {X}_{\mathrm {ord}}/\mathbb {F}_p$) be the $p$-divisible group corresponding to the covariant Dieudonné module $\mathbb {Z}_p^2$ with Frobenius $F$ acting by $b_{\mathrm {ss}}$ (respectively, $b_\mathrm {ord}$). Then $\mathbb {X}_{\mathrm {ss}}$ is a connected one-dimensional height-two $p$-divisible group, whereas $\mathbb {X}_{\mathrm {ord}}=\mu _{p^\infty } \times \mathbb {Q}_p/\mathbb {Z}_p$ is a one-dimensional height-two $p$-divisible group with non-trivial étale part. It follows from the classification of $p$-divisible groups by Dieudonné modules that, for any algebraically closed $\kappa /\mathbb {F}_p$, every height-two one-dimensional $p$-divisible group over $\kappa$ is quasi-isogenous/isomorphicFootnote ⁷ to exactly one of $\mathbb {X}_{\mathrm {ord}, \kappa }$ and $\mathbb {X}_{\mathrm {ss}, \kappa }$.

In particular, this applies to $E[p^\infty ]$ for $E/\kappa$ an elliptic curve. It thus makes sense, for any $K^p$ sufficiently small, to define the supersingular (respectively, ordinary) locus $\mathfrak {Y}_{K^p,\mathbb {F}_p}^\mathrm {ss}$ (respectively, $\mathfrak {Y}_{K^p,\mathbb {F}_p}^\mathrm {ord}$) in $\mathfrak {Y}_{K^p,\mathbb {F}_p}$ as the locus whose geometric points are such that the $p$-divisible group of universal elliptic curve is quasi-isogenous/isomorphic to $\mathbb {X}_\mathrm {ss}$ (respectively, $\mathbb {X}_\mathrm {ord}$). We then have

\[ \mathfrak{Y}_{K^p, \mathbb{F}_p} = \mathfrak{Y}_{K^p,\mathbb{F}_p}^\mathrm{ord} \sqcup \mathfrak{Y}_{K^p,\mathbb{F}_p}^\mathrm{ss} \]

and it can be shown that the supersingular locus is closed (and, thus, the ordinary locus is open). In fact, a local computation as in [Reference Katz and MazurKM85, Theorem 12.4.3] gives the following result.

We write also

\[ \mathfrak{X}_{K^p,\mathbb{F}_p}^\mathrm{ss} = \mathfrak{Y}_{K^p,\mathbb{F}_p}^\mathrm{ss} = V(\mathrm{Ha}) \quad\text{and}\quad \mathfrak{X}_{K^p,\mathbb{F}_p}^\mathrm{ord}=\mathfrak{X}_{K^p,\mathbb{F}_p} \backslash \mathfrak{X}_{K^p,\mathbb{F}_p}^\mathrm{ss}. \]

3.4 Ordinary Igusa varieties, mod $p$, and $p$-adic modular forms

Following Katz [Reference KatzKat75a], we define mod $p$ and $p$-adic modular forms as functions on certain moduli problems. This has the advantage of rendering group actions transparent, and mirrors our approach to the supersingular Igusa variety and quaternionic $p$-adic automorphic forms. Functions on these moduli problems have $q$-expansions, and the connection with the completion of $q$-expansions of classical modular forms as in Serre's [Reference SerreSer73] perspective then comes from evaluation along a canonical trivialization of $\omega$ and the $q$-expansion principle.

We begin by treating mod $p$ modular forms.

Definition 3.4.1 (The $\mu _p$-Igusa moduli problem)

Let $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$ be a closed subgroup. Let $\mathrm {Ig}^{\mu _p}_{K^p}$ be the functor on $\mathbb {F}_{p}$-algebras

\[ \mathrm{Ig}^{\mathrm{ord}}_{K^p,\mu_p}: R \mapsto \{ (E, \varphi_p, \mathcal{K}^p) \} /\sim \]

sending $R/\mathbb {F}_p$ to the set of equivalence classes of triples $(E, \varphi _p, \mathcal {K}^p)$ where:

1. (i) $E/R$ is an elliptic curve;

2. (ii) $\varphi _p: \mu _p \xrightarrow {\sim } \hat {E}[p]$ is an isomorphism;

3. (iii) $\mathcal {K}^p \subset \underline {\mathrm {Isom}}( (\underline {\mathbb {A}_f^{(p)}})^2, V_{\mathbb {A}_f^{(p)}}(E))$ is a $K^p$-torsor;

4. (iv) the relation $\sim$ is defined by $(E, \mathcal {K}^p) \sim (E', {\mathcal {K}^p}')$ if there is a prime-to-$p$ quasi-isogeny $q: E \rightarrow E'$ such that $q\circ \varphi _p = \varphi _p'$ and $q(\mathcal {K}^p) = {\mathcal {K}^p}'$.

As usual, when $K^p=\{e\}$ we drop it from the notation. For $K^p$ sufficiently small, the functor $\mathrm {Ig}^{\mathrm {ord}}_{K^p,\mu _p}$ is a finite étale $(\mathbb {Z}/p\mathbb {Z})^\times$ cover of $\mathfrak {Y}^{\mathrm {ord}}_{K^p, \mathbb {F}_p}$, where $(\mathbb {Z}/p\mathbb {Z})^\times$ acts by precomposition with $\varphi _p$. A direct computation on the Tate curve, whose formal group is canonically $\widehat {\mathbb {G}}_m$, shows the cover is unramified at the cusps and, thus, extends canonically to a finite étale $(\mathbb {Z}/p\mathbb {Z})^\times$-cover $\mathrm {Ig}^{\mathrm {ord},c}_{K^p,\mu _p}$ of $\mathfrak {X}^{\mathrm {ord}}_{K^p, \mathbb {F}_p}$.

The pullback $(\varphi _p^{-1})^*({dt}/{t})$ of the invariant differential ${dt}/{t}$ on $\mu _p=\operatorname {Spec}\!\mathbb {F}_p[t]/(t^p-1)$ is $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant trivialization of $\omega$ on $\mathrm {Ig}^{\mathrm {ord}}_{\mu _p}$ and extends to a trivialization over $\mathrm {Ig}^{\mathrm {ord}, c}_{\mu _p}$. In particular, we can use it to evaluate modular forms to functions on $\mathrm {Ig}^{\mathrm {ord}, c}_{\mu _p}$ and we obtain the following version of a well-known result (cf., e.g., [Reference GrossGros90]).

Lemma 3.4.3 Evaluation along $(\varphi _p^{-1})^*({dt}/{t})$ induces a $(\mathbb {Z}/p\mathbb {Z})^\times \times \mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant isomorphism of rings

\[ \biggl( \bigoplus_{k \geq 0} M_{k,\mathbb{F}_p} \biggr) / (\mathrm{Ha} - 1) \cong \mathcal{M}_{\mathbb{F}_p}, \]

where $(\mathbb {Z}/p\mathbb {Z})^\times$ acts on $M_{k,\mathbb {F}_p}$ by the character $z \mapsto z^{k}$.

Proof. First note that the evaluation map on $M_{k,\mathbb {F}_p}$ factors through $H^0(\mathfrak {X}^\mathrm {ord}_{\mathbb {F}_p},\omega ^k)$. One computes that $((\varphi _p^{-1})^*({dt}/{t}))^{p-1}=\mathrm {Ha}$, so that the evaluation map factors as

\[ \bigoplus_{k\geq 0} M_{k,\mathbb{F}_p} \rightarrow \bigoplus_{k=0}^{p-2} H^0(\mathfrak{X}^\mathrm{ord}_{\mathbb{F}_p},\omega^k) \xrightarrow{\sim} H^0(\mathrm{Ig}^{\mathrm{ord}, c}_{\\\mu_p}, \mathcal{O}) = \mathcal{M}_{\mathbb{F}_p}, \]

where the first arrow is restriction followed by division by $\mathrm {Ha}^{\lfloor k/(p-1)\rfloor }$. The second arrow is an isomorphism because we can decompose $H^0(\mathrm {Ig}^{\mathrm {ord}, c}_{\mu _p}, \mathcal {O})$ according to the characters of $(\mathbb {Z}/p\mathbb {Z})^\times$ in $\mathbb {F}_p$ and if $f$ is in the character space for $z^{k}$, then $f( (\varphi _p^{-1})^*({dt}/{t}))^k$ is invariant under $(\mathbb {Z}/p\mathbb {Z})^\times$ thus descends to a section over $\mathfrak {X}^\mathrm {ord}_{\mathbb {F}_p}$.

The first map is clearly surjective because multiplying by a sufficient power of $\mathrm {Ha}$ will clear any poles along the supersingular locus. On the other hand, any element in the kernel can be multiplied by $1+\mathrm {Ha} + \mathrm {Ha}^2 + \mathrm {Ha}^3 + \cdots$ (the condition of being in the kernel, when written out, guarantees that this product is zero in sufficiently large degree) to show that it is, in fact, in the ideal generated by $(1-\mathrm {Ha})$.

We now define $p$-adic modular forms.

Definition 3.4.5 (The Katz–Igusa moduli problem)

Let $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$ be a closed subgroup. Let $\mathrm {Ig}^{\mathrm {ord}}_{K^p, \widehat {\mathbb {G}}_m}$ be the functor on $\mathrm {Nilp}_{\mathbb {Z}_p}$

\[ \mathrm{Ig}^{\mathrm{ord}}_{K^p, \widehat{\mathbb{G}}_m}: R \mapsto \{(E, \varphi_p, \mathcal{K}^p)\} / \sim \]

sending $R/\mathbb {Z}_p$ to the equivalence classes of triples $(E, \varphi _p, \mathcal {K}^p)$ such that:

1. (i) $E/R$ is an elliptic curve;

2. (ii) $\varphi _p: \widehat {\mathbb {G}}_m \xrightarrow {\sim } \widehat {E}$ is an isomorphism of formal groups;

3. (iii) $\mathcal {K}^p \subset \underline {\mathrm {Isom}}( (\underline {\mathbb {A}_f^{(p)}})^2, V_{\mathbb {A}_f^{(p)}}(E) )$ is a $K^p$-torsor; and

4. (iv) the relation $\sim$ is defined by $(E, \varphi _p, \mathcal {K}^p) \sim (E', \varphi _p', {\mathcal {K}^p}')$ if there is a prime-to-$p$ quasi-isogeny $q: E \rightarrow E'$ such that $q\circ \varphi _p = \varphi _p'$ and $q(\mathcal {K}^p)=\mathcal {K}^p$.

As usual, when $K^p=\{e\}$ we drop it from the notation. The main source is [Reference KatzKat75a], which works with the geometrically connected variant with full level $n$ structure. As in [Reference KatzKat75a], for $K^p$ sufficiently small, $\mathrm {Ig}^{\mathrm {ord}}_{K^p, \widehat {\mathbb {G}}_m}$ is represented by an affine $p$-adic formal scheme, a pro-étale $\mathbb {Z}_p^\times$-torsor over the formal ordinary locus $\mathfrak {Y}_{K^p}^{\wedge, \mathrm {ord}}/\mathrm {Spf}\mathbb {Z}_p$; here, the action of $\underline {\mathbb {Z}_p^\times }$ is by precomposition with $\varphi _p$. For the same reason as the $\mu _p$-Igusa moduli problem, it is unramified at the cusps and thus extends canonically over $\mathfrak {X}^{\wedge,\mathrm {ord}}_{K^p} /\mathrm {Spf} \mathbb {Z}_p$ to an affine formal scheme $\mathrm {Ig}^{\mathrm {ord}, c}_{K^p, \widehat {\mathbb {G}}_m}$.

Definition 3.4.6 The space of $p$-adic modular forms is the unitary $\mathbb {Z}_p^\times \times \mathrm {GL}_2(\mathbb {A}_f^{(p)})$ representation on the $\mathbb {Q}_p$-Banach space

\[ \mathbb{V}_{\mathbb{Q}_p} := \mathbb{V}_{\mathbb{Z}_p}[1/p]\quad \text{for } \mathbb{V}_{\mathbb{Z}_p}:=H^0(\mathrm{Ig}^{\mathrm{ord},c}_{\widehat{\mathbb{G}}_m}, \mathcal{O}). \]

In the introduction, we used the notation $\mathcal {M}_{p-adic}$, which we do not employ further. We note also that $\mathbb {V}_{\mathbb {Z}_p}$ is $p$-torsion free.

The $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-representation $\mathbb {V}_{\mathbb {Q}_p}$ is not smooth, but one can check that the smooth vectors are dense and, for $K^p$ a sufficiently small closed subgroup, already at the integral level we have

\[ \mathbb{V}^{K^p}_{\mathbb{Z}_p} = H^0(\mathrm{Ig}^{\mathrm{ord}, c}_{K^p, \widehat{\mathbb{G}}_m}, \mathcal{O}). \]

The bundle $\omega$ is $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariantly trivialized over $\mathrm {Ig}^{\mathrm {ord}}_{\widehat {\mathbb {G}}_m}$ by $(\varphi _p^{-1})^*({dt}/{t})$ and this trivialization extends to $\mathrm {Ig}^{\mathrm {ord},c}_{\widehat {\mathbb {G}}_m}$. We can, thus, evaluate modular forms to elements of $\mathbb {V}_{\mathbb {Q}_p}$ and, as in [Reference KatzKat75a, especially Theorem 2.1], one obtains the following result.

Lemma 3.4.7 Evaluation on $(\varphi _p^{-1})^*({dt}/{t})$ induces a $\mathbb {Z}_p^\times \times \mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant injection

\[ \bigoplus_{k \geq 0} M_{k,\mathbb{Q}_p}^{\mathrm{GL}_2(\mathbb{Z}_p)} \hookrightarrow \mathbb{V}_{\mathbb{Q}_p}, \]

where we let $\mathbb {Z}_p^\times$ act by $z^{k}$ on $M_{k, \mathbb {Q}_p}$. The induced map on $K^p$-invariants has dense image for any compact open subgroup $K^p$.

Remark 3.4.8 If we write $\mathbb {V}_{\mathbb {F}_p}=\mathbb {V}_{\mathbb {Z}_p}/(p)$, then it is clear by comparing moduli problems that the invariants $\mathbb {V}_{\mathbb {F}_p}^{1+p\mathbb {Z}_p}$ represent $\mathrm {Ig}^{\mathrm {ord},c}_{\mu _p}$. The following obvious diagram comparing Lemmas 3.4.3 and 3.4.7 then commutes.

This diagram summarizes why in [Reference KatzKat75a] one must pass to divided congruences to see all of $\mathbb {V}$, which in our presentation corresponds to the fact that it is crucial to invert $p$ to obtain the density statement in Lemma 3.4.7.

Remark 3.4.9 For $K^p$ a sufficiently small compact open, if we fix a set of cusps $c_1, \ldots, c_m$, one in each connected component of $X_{\mathrm {GL}_2(\mathbb {Z}_p)K^p, \breve {\mathbb {Q}}_p}$, then we obtain $q$-expansion maps

\[ M_{k, \mathbb{Q}_p}^{\mathrm{GL}_2(\mathbb{Z}_p)K^p} \rightarrow \prod_{i=1}^{m} \breve{\mathbb{Z}}_p[[q]][1/p]. \]

A generalization of Serre's [Reference SerreSer73] original definition of $p$-adic modular forms of level $K^p$ would be to take the completion of the span of the images of these maps over all $k$. The $q$-expansion principle combined with Lemma 3.4.7 implies that this agrees with the definition given above (see [Reference KatzKat75a] for related discussions).

3.5 The supersingular Caraiani–Scholze Igusa variety

We now turn our attention to the main player in our story. Though it would be possible to work over $\mathbb {F}_p$, from here on out it will be cleaner and more convenient to work over $\overline {\mathbb {F}}_p$ (essentially because the endomorphisms of $\mathbb {X}_{\mathrm {ss}, \overline {\mathbb {F}}_p}$ are not all defined over $\mathbb {F}_p$).

Definition 3.5.1 (The supersingular Caraiani–Scholze Igusa moduli problem)

Let $K^p \leq \mathrm {GL}_2 (\mathbb {A}_f^{(p)})$ be a closed subgroup. Let $\mathrm {Ig}_{K^p}^\mathrm {ss}$ be the functor on $\overline {\mathbb {F}}_p$-algebras

\[ \mathrm{Ig}_{K^p}^{\mathrm{ss}}: R \mapsto \{(E, \varphi_p, \mathcal{K}^p)\}/\sim \]

sending $R/\overline {\mathbb {F}}_p$ to the set of equivalence classes of triples $(E, \varphi _p, \mathcal {K}^p )$ where:

1. (i) $E/R$ is an elliptic curve;

2. (ii) $\varphi _p: \mathbb {X}_{\mathrm {ss}, R} \otimes \mathbb {Q}_p \xrightarrow {\sim } E[p^\infty ] \otimes \mathbb {Q}_p$ is a quasi-isogeny;

3. (iii) $\mathcal {K}^p \subset \underline {\mathrm {Isom}}\bigl ((\underline {\mathbb {A}_f^{(p)}})^2, V_{\mathbb {A}_f^{(p)}}(E)\bigr )$ is a $K^p$-torsor; and

4. (iv) the relation $\sim$ is defined by $(E,\varphi _p, \mathcal {K}^p) \sim (E', \varphi _p', {\mathcal {K}^p}')$ if there is a quasi-isogeny $q: E \otimes \mathbb {Q} \xrightarrow {\sim } E'\otimes \mathbb {Q}$ such that

   \[ q(\mathcal{K}^p) = {\mathcal{K}^p}' \text{ and } q \circ \varphi_p = \varphi'_p. \]

As usual, when $K^p=\{e\}$ we drop it from the notation and write $\mathrm {Ig}^\mathrm {ss} := \mathrm {Ig}_{\{e\}}^\mathrm {ss}$. In this case, we also identify the prime-to-$p$ level data with the choice of an isomorphism $\varphi _{\mathbb {A}_f^{(p)}}: \underline {(\mathbb {A}_f^{(p)})^2} \rightarrow V_{\mathbb {A}_f^{(p)}}(E)$ as in Example 3.1.4.

This up to quasi-isogeny moduli problem is well-adapted for comparison with the quaternionic coset, as we show in § 3.7. To compare with modular curves, however, it is useful to also have an up to prime-to-$p$ quasi-isogeny moduli interpretation. Similarly to Lemma 3.1.5 (see also [Reference Caraiani and ScholzeCS17, Lemma 4.3.4]), we obtain the following result.

Lemma 3.5.2 For $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$ a closed subgroup, consider the functor

\[ \mathrm{Ig}^\mathrm{ss}_{K^p, \mathbb{X}_\mathrm{ss}}: R \mapsto \{(E, \varphi_p, \mathcal{K}^p)\}/\sim \]

sending $R/\overline {\mathbb {F}}_p$ to the set of equivalence classes of triples $(E, \varphi _p, \mathcal {K}^p)$ where:

1. (i) $E/R$ is an elliptic curve;

2. (ii) $\varphi _p: \mathbb {X}_{\mathrm {ss},R} \xrightarrow {\sim } E[p^\infty ]$ is an isomorphism;

3. (iii) $\mathcal {K}^p \subset \underline {\mathrm {Isom}}\bigl ((\underline {\mathbb {A}_f^{(p)}})^2, V_{\mathbb {A}_f^{(p)}}(E)\bigr )$ is a $K^p$-torsor; and

4. (iv) the relation $\sim$ is defined by $(E,\varphi _p, \mathcal {K}^p) \sim (E', \varphi _p', {\mathcal {K}^p}')$ if there is a prime-to- $p$ quasi-isogeny $q: E \otimes \mathbb {Z}_{(p)} \xrightarrow {\sim } E'\otimes \mathbb {Z}_{(p)}$ such that

   \[ q(\mathcal{K}^p) = {\mathcal{K}^p}' \quad \text{and} \quad q \circ \varphi_p = \varphi'_p. \]

The assignment

\[ (E, \varphi_p, \mathcal{K}^p) \mapsto (E, \varphi_p \otimes \mathbb{Q}_p, \mathcal{K}^p) \]

induces an isomorphism

\[ \mathrm{Ig}^\mathrm{ss}_{K^p, \mathbb{X}_\mathrm{ss}} \rightarrow \mathrm{Ig}^\mathrm{ss}_{K^p}. \]

We write $\mathcal {O}_p = \mathrm {End}(\mathbb {X}_{\mathrm {ss}, \overline {\mathbb {F}}_p})$ and $D_p := \mathcal {O}_p \otimes \mathbb {Q}_p$, so that $\mathcal {O}_p$ is the maximal order in $D_p$, a ramified quaternion algebra over $\mathbb {Q}_p$. Then $D_p$ acts on $\mathbb {X}_{\mathrm {ss}, \overline {\mathbb {F}}_p} \otimes \mathbb {Q}_p$, and there is a natural action of $D_p^\times$ on $\mathrm {Ig}^\mathrm {ss}_K$ by composition with $\varphi _p$. The action of $\mathcal {O}_p^\times \subset D_p^\times$ preserves the prime-to-$p$ moduli interpretation of Lemma 3.5.2.

In fact, what acts most naturally are the functors on $\overline {\mathbb {F}}_p$-algebras

\[ \underline{\mathrm{Aut}}(\mathbb{X}_{\mathrm{ss}, \overline{\mathbb{F}}_p}): R \mapsto \mathrm{Aut}(\mathbb{X}_{\mathrm{ss}, R}) \quad \text{and}\quad \underline{\mathrm{Aut}}(\mathbb{X}_{\mathrm{ss}, \overline{\mathbb{F}}_p} \otimes \mathbb{Q}_p): R \mapsto \mathrm{Aut} (\mathbb{X}_{\mathrm{ss}, R} \otimes \mathbb{Q}_p). \]

The following lemma says we have not missed anything.

3.6 Uniformization of the supersingular locus

Proposition 3.6.1 For $K^p$ a sufficiently small closed subgroup, the assignment

\[ (E, \varphi_p, \mathcal{K}^p ) \mapsto (E, \mathcal{K}^p ) \]

induces a $\underline {D_p^\times \times N_{\mathrm {GL}_2(\mathbb {A}_f^{(p)})}(K^p)}$-equivariant map

\[ \mathrm{unif}_{\mathbb{X}_\mathrm{ss}}: \mathrm{Ig}^\mathrm{ss}_{K^p}= \mathrm{Ig}^{\mathrm{ss}}_{K^p,\mathbb{X}_\mathrm{ss}} \rightarrow \mathfrak{Y}^\mathrm{ss}_{K^p, \overline{\mathbb{F}}_p}, \]

where $\underline {D_p^\times }$ acts on $\mathfrak {Y}^\mathrm {ss}_{K^p, \overline {\mathbb {F}}_p}$ by $\mathrm {Frob}^{v_p\circ \operatorname {Nrd}}$ (here $\mathrm {Frob}: \mathfrak {Y}_{K^p,\overline {\mathbb {F}}_p} \rightarrow \mathfrak {Y}_{K^p,\overline {\mathbb {F}}_p}$ is the relative Frobenius and $\operatorname {Nrd}: D_p \rightarrow \mathbb {Q}_p$ denotes the reduced norm). Moreover, $\mathrm {unif}_{\mathbb {X}_{\mathrm {ss}}}$ is a trivializable $\underline {\mathcal {O}_p^\times }$-torsor, and the maps compile to a $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant isomorphism of towers as the sufficiently small closed subgroup $K^p$ varies.

Proof. We first show that the map factors through $\mathfrak {Y}_{K^p,\overline {\mathbb {F}}_p}^\mathrm {ss}$: given a point

\[ (E, \varphi_p, \mathcal{K}^p) \in \mathrm{Ig}^\mathrm{ss}_{K^p,\mathbb{X}_\mathrm{ss}}(R), \]

we find $\mathrm {Ha}(E/R)$ is identically zero because $E[p^\infty ] \cong \mathbb {X}_{\mathrm {ss},R}$ and, as observed in § 3.2.2, the Hasse invariant only depends on $E[p^\infty ]$. However, $\mathrm {Ha}(E/R)$ is the pullback of $\mathrm {Ha}$ under the induced map $\operatorname {Spec}\!R \rightarrow \mathfrak {Y}_{K^p, \overline {\mathbb {F}}_p}$, so by Lemma 3.3.1 this induced map factors through $\mathfrak {Y}^\mathrm {ss}_{K^p, \overline {\mathbb {F}}_p}$, as desired.

It is also clear from the definitions of the moduli problems and the identification $\underline {\mathrm {Aut}}(\mathbb {X}_{\mathrm {ss}, \overline {\mathbb {F}}_p})=\underline {\mathcal {O}_p^\times }$ in Lemma 3.5.3 that the map is a quasi-torsor; to conclude it is a trivializable torsor it thus suffices to produce a section.

To obtain this, we observe that the $p$-divisible group of the universal elliptic curve over $\mathfrak {Y}_{K^p, \overline {\mathbb {F}}_p}^\mathrm {ss}$ is isomorphic to $\mathbb {X}_{\mathrm {ss}, \mathfrak {Y}_{K^p, \overline {\mathbb {F}}_p}^\mathrm {ss}}$: indeed, it is pulled back from $\mathfrak {Y}_{{K^p}', \overline {\mathbb {F}}_p}^\mathrm {ss}$ for any sufficiently small compact open subgroup ${K^p}' \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$ containing $K^p$, and $\mathfrak {Y}_{{K^p}', \overline {\mathbb {F}}_p}^\mathrm {ss}$ is just a finite union of $\overline {\mathbb {F}}_p$-points. We can choose an isomorphism at each of these points (because $\mathbb {X}_{\mathrm {ss}, \overline {\mathbb {F}}_p}$ is the unique up to isomorphism connected one-dimensional height-two $p$-divisible group over $\overline {\mathbb {F}}_p$), and then assemble these and pull back to $\mathfrak {Y}_{K^p}^\mathrm {ss}$ to obtain the desired isomorphism. This is exactly the data of a section.

It is left only to verify the action of $\underline {D_p^\times }/\underline {\mathcal {O}_p^\times }$ is as described on $\mathfrak {Y}_{K^p,\overline {\mathbb {F}}_p}^\mathrm {ss}$, and this is a direct computation from the definitions that we leave to the reader (we do not use this part of the statement in any of what follows).

As a consequence, one finds that for each sufficiently small $K^p$, $\mathrm {Ig}_{K^p}^\mathrm {ss}$ is a profinite set, thus, in particular, a perfect affine scheme over $\overline {\mathbb {F}}_p$. Our next goal is to identify this profinite set explicitly with a quaternionic coset.

3.7 Supersingular Igusa variety as quaternionic coset

We now identify $\mathrm {Ig}^\mathrm {ss}$ with a quaternionic coset using the orbit map for the action of $\underline {D_p^\times \times \mathrm {GL}_2(\mathbb {A}_f^{(p)})}$. To that end, fix a supersingular elliptic curve $E_0 / \overline {\mathbb {F}}_p$ and level structure to obtain

\[ x_0 = (E_0, \varphi_{p,0}, \varphi_{\mathbb{A}_f^{(p)},0}) \in \mathrm{Ig}^\mathrm{ss}({\overline{\mathbb{F}}_p}). \]

We write $D=\mathrm {End}(E_0) \otimes \mathbb {Q}$. As explained, e.g., in [Reference SilvermanSil92, Theorem V.3.1], this a quaternion algebra over $\mathbb {Q}$, and because it is non-split and acts faithfully on the $\ell$-adic Tate module for all $\ell \neq p$, it must be ramified exactly at $p$ and $\infty$.

By definition, $D^\times$ is identified with the self quasi-isogenies of $E_0 \otimes \mathbb {Q}$. The actions of $D$ on $E_0[p^\infty ]\otimes \mathbb {Q}_p$ and $V_{\mathbb {A}_f^{(p)}}(E)$, transported by $\varphi _{p,0}$ and $\varphi _{\mathbb {A}_f^{(p)},0}$, thus induce an identification

\[ D^\times(\mathbb{A}_f) = D^\times(\mathbb{Q}_p) \times D^\times(\mathbb{A}_f^{(p)})= D_p^\times \times \mathrm{GL}_2(\mathbb{A}_f^{(p)}). \]

In particular, we obtain a map $D^\times (\mathbb {Q}) \hookrightarrow D^\times (\mathbb {A}_f)$.

The action on the point $x_0$ induces an orbit map

(3.7.1)\begin{equation} \underline{D_p^\times \times \mathrm{GL}_2(\mathbb{A}_f^{(p)})} \rightarrow \mathrm{Ig}^\mathrm{ss}, \quad g \mapsto x_0 g, \end{equation}

and we show that the following holds.

Theorem 3.7.1 The orbit map ( 3.7.1) factors through a ${D^\times (\mathbb {A}_f)=D^\times (\mathbb {Q}_p) \times \mathrm {GL}_2(\mathbb {A}_f^{(p)})}$-equivariant isomorphism

\[ \underline{ D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f)} = \underline{D^\times(\mathbb{Q}) \backslash \big( D_p^\times \times \mathrm{GL}_2\big(\mathbb{A}_f^{(p)}\big) \big)} \xrightarrow{\sim} \mathrm{Ig}^{\mathrm{ss}}. \]

Before proving Theorem 3.7.1, it is helpful to recall the following basic structural result for quaternionic double cosets. We write $\mathcal {O} = \mathrm {End}(E_0)$, a maximal order in $D$, and consider the finite class set of right fractional ideals in $\mathcal {O}$

\[ D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f) / (\mathcal{O} \otimes \widehat{\mathbb{Z}})^\times = D^\times(\mathbb{Q}) \backslash D_p^\times \times \mathrm{GL}_2(\mathbb{A}_f^{(p)}) / \mathcal{O}_p^\times \times \mathrm{GL}_2(\widehat{\mathbb{Z}}^{(p)}). \]

We may fix a finite set of representatives for the class set $I$ and corresponding representatives $\gamma _\mathcal {I} \in D^\times (\mathbb {A}_f),\ \mathcal {I} \in I$, for the double cosets. The stabilizer of $\mathcal {I}$ for left multiplication in $D^\times (\mathbb {Q})$ is the units in a maximal order $\mathcal {O}_\mathcal {I}$, and we find

\[ D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f) \cong \bigsqcup_{\mathcal{I} \in I} \mathcal{O}_\mathcal{I}^\times \backslash \gamma_\mathcal{I} (\mathcal{O} \otimes \widehat{\mathbb{Z}})^\times = \bigsqcup_{\mathcal{I} \in I} \mathcal{O}_{\mathcal{I}}^\times \backslash \mathrm{Isom}(\mathcal{O} \otimes \widehat{\mathbb{Z}}, \mathcal{I} \otimes \widehat{\mathbb{Z}}),\]

where the isomorphisms on the right-hand side are of right $\mathcal {O} \otimes \widehat {\mathbb {Z}}$-modules. Because each group $\mathcal {O}_\mathcal {I}^\times$ is finite, if we replace $\mathcal {O} \otimes \widehat {\mathbb {Z}}$ with a small enough compact open subgroup $K \subset \mathcal {O} \otimes \widehat {\mathbb {Z}}^\times$, we obtain a finite set of representatives $\gamma _{\mathcal {I}, i}$ such that

(3.7.2)\begin{equation} D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f) \cong \bigsqcup_{\mathcal{I} \in I, i} \gamma_{\mathcal{I},i} K. \end{equation}

In other words, we obtain a topological splitting of the locally profinite set $D^\times (\mathbb {A}_f)$ as a product of a discrete set and a profinite set,

(3.7.3)\begin{equation} D^\times(\mathbb{A}_f) \cong D^\times(\mathbb{Q}) \times D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f) \end{equation}

compatible with the left action of $D^\times (\mathbb {Q})$ and the right action of $K$.

We also need to understand the quasi-isogenies of $E_0$ after arbitrary base change. To this end, we observe that, for any $S$, there is a natural map from $\underline {D^\times (\mathbb {Q})}(R)$ to $\underline {\mathrm {Aut}}(E_0\otimes \mathbb {Q})(R)=\mathrm {Aut}(E_{0,R} \otimes \mathbb {Q})$; indeed, because $D^\times (\mathbb {Q})$ is discrete, an element of $\underline {D^\times (\mathbb {Q})}(R)$ is a locally constant on $\operatorname {Spec}\!R$ choice of element in $D^\times (\mathbb {Q})$.

Proof of Theorem 3.7.1 By Lemma 3.7.2, we deduce that the orbit map factors as an injection on $R$-points

\[ \underline{D^\times(\mathbb{Q})}(R) \backslash \underline{D^\times(\mathbb{A}_f)}(R) \rightarrow \mathrm{Ig}^\mathrm{ss} (R). \]

From (3.7.3), we deduce

\[ \underline{D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f)} (R) = \underline{D^\times(\mathbb{Q})}(R) \backslash \underline{D^\times(\mathbb{A}_f)}(R), \]

and thus the orbit map factors through an injection

\[ \underline{D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f)} \hookrightarrow \mathrm{Ig}^{\mathrm{ss}}. \]

It remains to show the map is surjective. To do so, it suffices to show that the universal elliptic curve over $\mathrm {Ig}_\mathrm {ss}$ is quasi-isogenous to $E_{0,\mathrm {Ig}^\mathrm {ss}}$. However, the universal elliptic curve is the pullback from $Y^\mathrm {ss}_{\overline {\mathbb {F}}_p}$ along the map $\mathrm {unif}_{\mathbb {X}_\mathrm {ss}}$ of Proposition 3.6.1, and the statement follows as in the proof of Proposition 3.6.1 by reduction to a finite set of points at finite level and the fact that any two supersingular elliptic curves over ${\overline {\mathbb {F}}_p}$ are isogenous.

Corollary 3.7.4 For any sufficiently small closed subgroup $K^p\leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$, the map of Theorem 3.7.1 induces a $\underline {D_p^\times \times N_{\mathrm {GL}_2(\mathbb {A}_f^{(p)})}(K^p)}$-equivariant isomorphism

\[ \underline{D^\times(\mathbb{Q}) \backslash \big( D_p^\times \times \mathrm{GL}_2\big(\mathbb{A}_f^{(p)}\big) \big)/K^p } \xrightarrow{\sim} \mathrm{Ig}^{\mathrm{ss}}_{K^p} \]

and, combined with Proposition 3.6.1,

\[ \underline{D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f) / \big(\mathcal{O}_p^\times \times K^p\big)}\xrightarrow{\sim} \mathfrak{Y}_{K^p, \overline{\mathbb{F}}_p}^\mathrm{ss}. \]

These maps compile to $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant isomorphisms of towers as the closed subgroup $K^p$ varies over all sufficiently small closed subgroups.

4.1 Mod $p$ modular forms

We write $\mathcal {M}_{\overline {\mathbb {F}}_p}$ for the base change to $\overline {\mathbb {F}}_p$ of the space of mod $p$ modular forms of Definition 3.4.2. As explained in the proof of Lemma 3.4.3, the evaluation map from modular forms to mod $p$ modular forms factors through an isomorphism of $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-representations

(4.1.1)\begin{equation} \mathcal{M}_{\overline{\mathbb{F}}_p} = \bigoplus_{k=0}^{p-2} H^0(\mathfrak{X}_{\overline{\mathbb{F}}_p}^\mathrm{ord}, \omega^k). \end{equation}

We consider the increasing exhaustive filtration $F_i \mathcal {M}_{\overline {\mathbb {F}}_p}$ whose $i$th step consists of those sections on the right-hand side of (4.1.1) with poles of order $\leq i$ along $\mathfrak {X}_{\overline {\mathbb {F}}_p}^\mathrm {ss}$.

As in § 3.2, we also write

\[ M_{k, \overline{\mathbb{F}}_p} = H^0(\mathfrak{X}_{\overline{\mathbb{F}}_p}, \omega^k), \]

the admissible $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$ representation of weight $k$ modular forms over $\overline {\mathbb {F}}_p$. In particular, for $K^p$ sufficiently small, we have

\[ M_{k, \overline{\mathbb{F}}_p}^{K^p} = H^0(\mathfrak{X}_{K^p, \overline{\mathbb{F}}_p}, \omega^k). \]

From these definitions and Lemma 3.3.1, we find that multiplication by $\mathrm {Ha}^i$ gives a $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant isomorphism

(4.1.2)\begin{equation} F_i \mathcal{M}_{\overline{\mathbb{F}}_p} \xrightarrow{\sim} \bigoplus_{k=0}^{p-2} M_{k+(p-1)i,\overline{\mathbb{F}}_p}. \end{equation}

Thus, in particular, $F_i \mathcal {M}_{\overline {\mathbb {F}}_p}$ is an admissible $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-representation.

4.2 Evaluation of modular forms

We now explain how to evaluate modular forms to functions on $\mathrm {Ig}^\mathrm {ss}$. This works almost exactly as in the evaluation map for mod $p$ modular forms in Lemma 3.4.3, but we set things up here in a slightly more canonical language.

We write $\omega _{\mathbb {X}_\mathrm {ss}}=\operatorname {Lie}(\mathbb {X}_\mathrm {ss})^*$, equipped with the natural action of $\mathcal {O}_p^\times$ after base change to $\overline {\mathbb {F}}_p$. Using the isomorphism $\varphi _p$ in the prime-to-$p$ moduli interpretation (cf. Definition 3.5.1(ii)) and the uniformization in Proposition 3.6.1, we obtain a canonical $\mathcal {O}_p^\times \times \mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant isomorphism

\[ \omega_{\mathbb{X}_\mathrm{ss}} \otimes_{\mathbb{F}_p} \mathcal{O}_{\mathrm{Ig}^{\mathrm{ss}}} \xrightarrow{\sim} \mathrm{unif}_{\mathbb{X}_\mathrm{ss}}^* \omega. \]

In particular, we obtain a $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant isomorphism

\[ H^0\bigl(\mathfrak{X}_{\overline{\mathbb{F}}_p}^\mathrm{ss}, \omega^k) \xrightarrow{\sim} \mathrm{Hom}_{\mathcal{O}_p^\times}( \operatorname{Lie}(\mathbb{X}_{\mathrm{ss},\overline{\mathbb{F}}_p})^k, H^0(\mathrm{Ig}^\mathrm{ss}, \mathcal{O}) \bigr). \]

Note that, by the definition of $\mathbb {X}_\mathrm {ss}$ in terms of the Dieudonné module given in § 3.5, we have an identification of $\operatorname {Lie} \mathbb {X}_{\mathrm {ss}}$ with $\mathbb {F}_p^2/\langle (1, 0) \rangle$, and of $\mathcal {O}_p$ with the $\sigma$-centralizer of $b_\mathrm {ss}$ in $M_2(\breve {\mathbb {Z}}_p)$. In particular, the image of $(0,1)$ gives a basis element of $\operatorname {Lie} \mathbb {X}_\mathrm {ss}$, and a direct computation shows $\mathcal {O}_p^\times$ acts on $\operatorname {Lie} \mathbb {X}_{\mathrm {ss},\overline {\mathbb {F}}_p}={\overline {\mathbb {F}}_p}$ through a surjective character $\epsilon : \mathcal {O}_p^\times \rightarrow \mathbb {F}_{p^2}^\times$ whose kernel is a pro-$p$ group we write as $\mathcal {N}_p$. Combining with Theorem 3.7.1, we obtain a $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-equivariant isomorphism

(4.2.1)\begin{equation} H^0(\mathfrak{X}_{\overline{\mathbb{F}}_p}^\mathrm{ss}, \omega^k) \xrightarrow{\sim} \mathrm{Hom}_{\mathcal{O}_p^\times}\big( \epsilon^k, \mathrm{Cont}\big(D^\times(\mathbb{Q}) \backslash D_p \times \mathrm{GL}_2(\mathbb{A}_f^{(p)}) / \mathcal{N}_p, {\overline{\mathbb{F}}_p} \big) \big). \end{equation}

Then, evaluating homomorphisms in the right-hand side of (4.2.1) on $1 \in {\overline {\mathbb {F}}_p}$ and passing to $K^p$-invariants, we obtain Serre's map, a Hecke equivariant isomorphism

(4.2.2)\begin{equation} H^0(\mathfrak{X}_{\overline{\mathbb{F}}_p}^\mathrm{ss}, \omega^k) \xrightarrow{\sim} \mathcal{A}^{\mathcal{N}_p K^p}_{{\overline{\mathbb{F}}_p}}[\epsilon^k]. \end{equation}

4.3 Hecke algebras and generalized eigenspaces

If $L$ is a field and $V$ is a finite-dimensional vector space over $L$, then for any commutative $L$-algebra $T$ acting on $V$, we have a decomposition into generalized eigenspaces

\[ V = \bigoplus V_{\mathfrak{m}}, \]

where $\mathfrak {m}$ runs over maximal ideals of $T$ with residue field a finite extension of $L$ and $V_\mathfrak {m}$ is the sub-module of $\mathfrak {m}$-torsion elements. In particular, if $V_\mathfrak {m} \neq 0$, the eigenspace $V[\mathfrak {m}]$ consisting of elements annihilated by $\mathfrak {m}$ is non-empty.

The same result applies more generally to $V$ an increasing union of finite-dimensional vector spaces with $T$-action. In particular, if $W$ is an increasing union of admissible representations of a locally profinite group $G$, $K$ is a compact open subgroup of $G$, and $T$ is a commutative subalgebra of the abstract Hecke algebra $L[K\backslash G/K]$, then the action of $T$ on the invariants $W^K$ admits a decomposition into generalized eigenspaces.

This formalism gives a decomposition into generalized eigenspaces for the action of any commutative subalgebra

\[ \mathbb{T}' \subset \overline{\mathbb{F}}_p[K^p\backslash\mathrm{GL}_2(\mathbb{A}_f^{(p)}/K^p] = \mathbb{T}^\mathrm{abs}_{K^p} \otimes \overline{\mathbb{F}}_p, \]

on $M_{k, \overline {\mathbb {F}}_p}^{K^p}$, $\mathcal {A}^{K^p}_{\overline {\mathbb {F}}_p}$, and $\mathcal {M}_{\overline {\mathbb {F}}_p}^{K^p}$. For $M_{k,\overline {\mathbb {F}}_p}^{K^p}$ this is immediate because $M_{k, \overline {\mathbb {F}}_p}$ is admissible. For $\mathcal {A}^{K^p}_{\overline {\mathbb {F}}_p}$ it follows because $\mathcal {A}$ is a colimit of admissible $\mathrm {GL}_2(\mathbb {A}_f^{(p)})$-representations by taking invariants under compact open subgroups of $D_p^\times$. For $\mathcal {M}_{\overline {\mathbb {F}}_p}^{K^p}$ it follows using the filtration by the admissible representations $F_i \mathcal {M}_{\overline {\mathbb {F}}_p}$.

4.4 Spectral decompositions

We now compare the spectral decompositions provided by the previous section in order to prove Theorem 1.1.1. The comparison of $\mathcal {M}_{\overline {\mathbb {F}}_p}^{K^p}$ and $\mathcal {A}^{K^p}_{\overline {\mathbb {F}}_p}$ is mediated through comparisons of each with $\bigoplus _{k\geq 0}M_{k, \overline {\mathbb {F}}_p}^{K^p}$.

We first treat the simpler case of $\mathcal {M}_{\overline {\mathbb {F}}_p}^{K^p}$.

Lemma 4.4.1 For $\mathfrak {m}$ a maximal ideal of $\mathbb {T}'$,

\[ \big( \mathcal{M}^{K^p}_{\overline{\mathbb{F}}_p} \big)_\mathfrak{m} \neq 0\iff \bigoplus_{k\geq 0}\big(M_{k, \overline{\mathbb{F}}_p}^{K^p}\big)_\mathfrak{m} \neq 0. \]