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The spectral p-adic Jacquet–Langlands correspondence and a question of Serre

Published online by Cambridge University Press:  11 April 2022

Sean Howe*
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT84105, USAsean.howe@utah.edu
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Abstract

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Compositio Mathematica is © Foundation Compositio Mathematica.
Copyright
© 2022 The Author(s)

1. Introduction

Let $p$ be a prime and let $D/\mathbb {Q}$ be the (unique up to isomorphism) quaternion algebra ramified at $p$ and $\infty$. Let $\mathbb {A}$ denote the adèles of $\mathbb {Q}$, $\mathbb {A}_f$ the finite adèles, and $\mathbb {A}_f^{(p)}$ the finite prime-to-$p$ adèles. Let $K^p \subset D^\times (\mathbb {A}_f^{(p)})$ be a compact open subgroup. For $R$ a topological ring (e.g. $\mathbb {C}$, ${\overline {\mathbb {F}}_p}$, $\mathbb {Q}_p$, or $\mathbb {C}_p$), we consider the space of continuous $p$-adic automorphic forms on $D^\times$ with coefficients in $R$ and prime-to-$p$ level $K^p$,

\[ \mathcal{A}_{R}^{K^p} := \mathrm{Cont}(D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}) / K^p , R). \]

For $R$ totally disconnected (e.g. ${\overline {\mathbb {F}}_p}$, $\mathbb {Q}_p$, or $\mathbb {C}_p$), the archimedean component can be removed, and we have an identification

\[ \mathcal{A}_R^{K^p} = \mathrm{Cont}(D^\times(\mathbb{Q}) \backslash D^\times(\mathbb{A}_f) / K^p , R). \]

Note that $D^\times (\mathbb {Q}) \backslash D^\times (\mathbb {A}_f) / K^p$ is a profinite set. Moreover, by choosing coset representatives, it can be identified with a finite disjoint union of compact open subgroups of $D^\times (\mathbb {Q}_p)$, so that it is essentially a $p$-adic object.

The space $\mathcal {A}_{R}^{K^p}$ admits an action of the abstract double-coset Hecke algebra

\[ \mathbb{T}_{\mathrm{abs}} := \mathbb{Z}[ K^p \backslash D^\times(\mathbb{A}_f^{(p)}) / K^p], \]

and a commuting action of $D^\times (\mathbb {Q}_p)$. In this work, we study the spectral decomposition of $\mathcal {A}_{R}^{K_p}$ under the action of $\mathbb {T}_{\mathrm {abs}}$.

The classical Jacquet–Langlands correspondence [Reference Jacquet and LanglandsJL70], proved using analytic techniques, implies that, up to twisting, the eigensystems for $\mathbb {T}_{\mathrm {abs}}$ acting on $\mathcal {A}_{\mathbb {C}}^{K^p}$ are a strict subset of those appearing in classical complex modular forms. The eigensystem attached to a cuspidal modular form appears on the quaternionic side if and only if the corresponding automorphic representation of $\mathrm {GL}_2$ is discrete series at $p$.

On the other hand, arguing with the geometry of mod $p$ modular curves, Serre [Reference SerreSer96] showed that the eigensystems arising in $\mathcal {A}_{{\overline {\mathbb {F}}_p}}^{K^p}$ are the same as those appearing in the space of mod $p$ modular forms (see Theorem 1.1.1 for a slight refinement of Serre's result). In particular, the gaps in the Jacquet–Langlands correspondence over $\mathbb {C}$ disappear when working mod $p$.

The main result of this work is a natural lift of Serre's result to $\mathbb {Q}_p$: we use the geometry of the perfectoid modular curve at infinite level to show that the completed Hecke algebra of $\mathcal {A}_{\mathbb {Q}_p}^{K^p}$ is isomorphic to the completed Hecke algebra of $p$-adic modular forms (see Theorem A for a precise statement).

Theorem A is compatible with the classical Jacquet–Langlands correspondence: the eigensystems appearing in classical complex quaternionic automorphic forms can be identified with the eigensystems appearing in $\mathcal {A}_{\mathbb {Q}_p}^{K_p}$ such that the corresponding eigenspace contains a vector which, up to a twist, transforms via an algebraic representation of $D^\times (\mathbb {Q}_p)$ after restriction to a sufficiently small compact open subgroup. Thus, Theorem A can be interpreted as saying that there is a $p$-adic Jacquet–Langlands correspondence that fills in the gaps in the classical Jacquet–Langlands correspondence. As our proof of the $p$-adic correspondence is independent of the classical correspondence, we also obtain a new proof that Galois representations can be attached to quaternionic automorphic forms (Corollary B).

Both Theorems 1.1.1 and A are purely spectral Jacquet–Langlands correspondences, in the sense that they compare spectral information for a family of prime-to-$p$ Hecke operators acting on two different spaces but say little else relating the structure of these spaces; in particular, we make no attempt here to describe the local $D^\times (\mathbb {Q}_p)$-representation appearing in a fixed Hecke eigenspace. Nevertheless, some of the methods employed in the proofs of Theorems 1.1.1 and A can be used provide significant information about these local representations, and we plan to return to this in future work (see § 1.3 for further discussion).

1.1 Serre's spectral mod $p$ Jacquet–Langlands correspondence

Before discussing our results and techniques further, we take a detour to give a precise statement of Serre's [Reference SerreSer96] mod $p$ correspondence.

If we fix an isomorphism

\[ D^\times(\mathbb{A}_f^{(p)})\cong \mathrm{GL}_2\big(\mathbb{A}_f^{(p)}\big), \]

then we obtain an action of the Hecke algebra $\mathbb {T}_\mathrm {abs}$ on the space $\mathcal {M}_{{\overline {\mathbb {F}}_p}}^{K^p}$ of mod $p$ modular forms of prime-to-$p$ level $K^p$. In a 1987 letter to Tate, Serre [Reference SerreSer96] proved a mod $p$ Jacquet–Langlands correspondence comparing the spectral decompositions of $\mathcal {A}^{K^p}_{{\overline {\mathbb {F}}_p}}$ and $\mathcal {M}_{{\overline {\mathbb {F}}_p}}^{K^p}$. We state below a slight strengthening of his result, which follows essentially from Serre's proof.Footnote 1 First, some notation.

Suppose $\mathbb {T}' \subset \mathbb {T}_{\mathrm {abs}}$ is a commutative sub-algebra and $\chi : \mathbb {T}' \rightarrow {\overline {\mathbb {F}}_p}$ is a character. Then, if $\mathbb {T}'$ acts on an ${\overline {\mathbb {F}}_p}$-vector space $V$, we may consider the $\chi$-eigenspace $V[\chi ]$. If we write $\mathfrak {m}_\chi := \ker \chi$, we may also consider the generalized $\chi$-eigenspace $V_{\mathfrak {m}_\chi }$ (that is, the subset of elements killed by a power of $\mathfrak {m}_\chi$).

Theorem 1.1.1 (Serre)

Let $\mathbb {T}' \subset \mathbb {T}_{\mathrm {abs}}$ be a commutative sub-algebra. Then, there is a finite collection of characters $\chi _i: \mathbb {T}' \rightarrow {\overline {\mathbb {F}}_p}$ with kernels $\mathfrak {m}_i$ such that:

  1. (i) for each $i$, $\big (\mathcal {A}^{K^p}_{{\overline {\mathbb {F}}_p}}\big )_{\mathfrak {m}_i}$ and $\big (\mathcal {M}^{K^p}_{{\overline {\mathbb {F}}_p}}\big )_{\mathfrak {m}_i}$ are non-zero; in particular

    \[ \mathcal{A}^{K^p}_{{\overline{\mathbb{F}}_p}}[\chi_i] \neq 0 \quad \text{and} \quad \mathcal{M}^{K^p}_{{\overline{\mathbb{F}}_p}}[\chi_i] \neq 0; \]
  2. (ii) there are direct sum decompositions

    \[ \mathcal{A}^{K^p}_{{\overline{\mathbb{F}}_p}} = \bigoplus_i \big( \mathcal{A}^{K^p}_{{\overline{\mathbb{F}}_p}} \big)_{\mathfrak{m}_{\chi_i}} \quad \text{and} \quad \mathcal{M}_{{\overline{\mathbb{F}}_p}}^{K^p} = \bigoplus_i \big( \mathcal{M}_{{\overline{\mathbb{F}}_p}}^{K^p}\big)_{\mathfrak{m}_{\chi_i}}. \]

In other words, the Hecke eigensystems appearing in mod $p$ quaternionic automorphic forms are precisely those appearing in mod $p$ modular forms. This stands in contrast to the classical Jacquet–Langlands correspondence, where the eigensystems appearing in quaternionic forms are a strict subset of those appearing in modular forms. The following example gives a concrete illustration.

Example 1.1.2 The discriminant form, represented by the Ramanujan series

\[ \Delta(q)=q \prod_{n \geq 1}(1-q^n)^{24} = \sum \tau(n) q^n, \]

is a weight 12 level-one cuspidal eigenform whose corresponding automorphic representation is principal series at every prime $p$. Thus, the classical Jacquet–Langlands correspondence says that its Hecke eigensystem, encoded by the coefficients $\tau (\ell )$ for $\ell$ prime, does not appear in the space of classical automorphic forms on $D^\times$ for any prime $p$ (recall $p$ appears in the definition of $D^\times$). By contrast, Theorem 1.1.1 shows that the coefficients $\tau (\ell ) \mod p$ for $\ell \neq p$ are remembered by a mod $p$ quaternionic automorphic form on $D^\times$. A similar phenomenon occurs in our $p$-adic correspondence, which remembers the numbers $\tau (\ell )$ on the nose.

1.2 A spectral $p$-adic Jacquet–Langlands correspondence

1.2.1 Serre's question

Serre ended his letter to Tate with a list of questions inspired by the mod $p$ Jacquet–Langlands correspondence. One of these suggests an analogous study relating $\mathcal {A}_{\overline {\mathbb {Q}_p}}^{K^p}$ to $p$-adic modular forms.

Analogues $p$-adiques. Au lieu de regarder les fonctions localement constantes sur $D_\mathbb {A}^\times / D_{\mathbb {Q}}^\times$ á valeurs dans $\mathbb {C}$, il serait plus amusant de regarder celles à valeurs dans $\overline {\mathbb {Q}}_p$. Si l'on décompose $\mathbb {A}$ en $\mathbb {Q}_p \times \mathbb {A}'$, on leur imposerait d’être localement constantes par rapport à la variable dans $D_{\mathbb {A}'}$ et d’être continues (ou analytiques, ou davantage) par rapport à la variable dans $D_p$…Y aurait-il des représentations galoisiennes $p$-adiques associées a de telles fonctions, supposées fonctions propres des opérateurs de Hecke? Peut-on interpréter les constructions de Hida (et Mazur) dans un tel style? Je n'en ai aucune idée. (Serre [Reference SerreSer96, paragraph (26)].)

Our main result, Theorem A, shows that the answers to Serre's questions are, largely, yes. In particular, Theorem A implies that Galois representations can be attached to $p$-adic quaternionic eigenforms (Corollary B).

1.2.2 A homeomorphism of completed Hecke algebras

The space $\mathcal {A}^{K^p}_{\mathbb {Q}_p}$ of $p$-adic quaternionic automorphic forms is a $\mathbb {Q}_p$-Banach space with respect to the sup norm and the action of $\mathbb {T}_\mathrm {abs}$ is by bounded linear operators. For any subalgebra $\mathbb {T}'\subset \mathbb {T}_\mathrm {abs}$ we thus obtain a completed Hecke algebra ${\mathbb {T}'}^\wedge _{\mathcal {A}^{K^p}_{\mathbb {Q}_p}}$ by taking the closure of the image of $\mathbb {T}'$ in the algebra of bounded linear operators on $\mathcal {A}^{K^p}_{\mathbb {Q}_p}$ (equipped with the topology of pointwise convergence; see § 2.4 for details). It is a topological $\mathbb {Z}_p$-algebra.

As in the mod $p$ case, we would like to compare the Hecke action on $\mathcal {A}^{K^p}_{\mathbb {Q}_p}$ to a Hecke action on a space of modular forms; in this case, we do so by comparing completed Hecke algebras. To that end: Serre [Reference SerreSer73] constructed natural spaces of $p$-adic modular forms by completing spaces of classical modular forms for the $p$-adic topology on $q$-expansions (these spaces were then interpreted geometrically by Katz [Reference KatzKat75a, Reference KatzKat75b]). In particular, one obtains a natural $\mathbb {Q}_p$-Banach space $\mathcal {M}_{p\text {-}\mathrm {adic}}^{K^p}$ of $p$-adic modular forms of prime-to-$p$ level $K^p$ equipped with an action of $\mathbb {T}_\mathrm {abs}$ by bounded linear operators and, thus, a completed Hecke algebra ${\mathbb {T}'}^\wedge _{\mathcal {M}^{K^p}_{p\text {-}\mathrm {adic}}}$. Our main result is as follows.

Theorem A For any sub-algebra $\mathbb {T}' \subset \mathbb {T}_\mathrm {abs}$, the identity map $\mathbb {T}' \rightarrow \mathbb {T}'$ extends to a canonical isomorphism of topological $\mathbb {Z}_p$-algebras

\[ {\mathbb{T}'}^\wedge_{\mathcal{A}^{K^p}_{\mathbb{Q}_p}} = {\mathbb{T}'}^\wedge_{\mathcal{M}^{K^p}_{p\text{-}\mathrm{adic}}}. \]

Theorem A implies Theorem 1.1.1 (as essentially explained in § 4.5; the point is that the maximal ideals in Theorem 1.1.1 correspond to the open maximal ideals in the corresponding completed Hecke algebras), and gives a natural lift to characteristic zero suitable, e.g., for the construction of Galois representations. Our proof lifts Serre's approach via the mod $p$ geometry of modular curves to characteristic zero by using the $p$-adic geometry of infinite level modular curves.

Remark 1.2.3 The completed Hecke algebras do not change if we replace $\mathbb {Q}_p$ with a finite extension, or even $\mathbb {C}_p$ (and, indeed, this invariance under base change plays an important role in our proof). Thus, although Serre in his letter quoted above suggests the study of $\mathcal {A}^{K^p}_{\overline {\mathbb {Q}}_p}$, it is natural in our setup to work over $\mathbb {Q}_p$. In particular, an eigenform in $\mathcal {A}^{K^p}_{\overline {\mathbb {Q}}_p}$ will still give rise to a $\overline {\mathbb {Q}}_p$-valued character of ${\mathbb {T}'}^\wedge _{\mathcal {A}^{K^p}_{\mathbb {Q}_p}}$.

1.2.4 Completed cohomology

By a result of Emerton [Reference EmertonEme11] (building on work by Hida), ${{\mathbb {T}}'}^\wedge _{\mathcal {M}^{K^p}_{p\text {-}\mathrm {adic}}}$ is equal to the completed Hecke algebra of $\mathbb {T}'$ acting on the completed cohomology of modular curves. On the other hand, $\mathcal {A}^{K^p}_{\mathbb {Q}_p}$ is the completed cohomology at level $K^p$ for $D^\times$. Hence, we also obtain a homeomorphism between the completed Hecke algebras for the completed cohomology of $\mathrm {GL}_2$ and $D^\times$. In fact, our proof passes first through this equivalence then uses the result of Emerton, which we establish more carefully along with some other identifications in § 5.7.

1.2.5 Galois representations

Let $\mathbb {T}_{\mathrm {tame}} \subset \mathbb {T}_\mathrm {abs}$ be the tame Hecke algebra of level $K^p$, i.e. the commutative sub-algebra generated by the Hecke operators at $\ell$ for primes $\ell \neq p$ at which $K^p$ factors as $K^{p,\ell }K_\ell$ for $K^{p,\ell } \subset D^\times (\mathbb {A}_f^{(pl)})$ and $K_\ell \subset D^\times (\mathbb {Q}_l)$ a maximal compact subgroup. For each such $\ell$ we write $T_\ell$ for the standard Hecke operator. Combining Theorem A with known results for ${\mathbb {T}'}^\wedge _{\mathcal {M}^{K^p}_{p\text {-}\mathrm {adic}}}$ gives the following result.

Corollary B If $\chi : \mathbb {T}^\wedge _{\mathrm {tame}, \mathcal {A}^{K^p}_{\mathbb {Q}_p}} \rightarrow \mathbb {C}_p$ is a continuous character then there exists a unique semisimple continuous representation

\[ \rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}_2(\mathbb{C}_p) \]

unramified at $\ell$ as above and such that $\mathrm {Tr}(\rho (\mathrm {Frob}_\ell ))=\chi (T_\ell )$.

One can obtain such a $\chi$ from a quaternionic eigenform as in Remark 1.2.3, and thus Corollary B attaches Galois representations to these eigenforms.

1.2.6 Work of Emerton

Corollary B can also be deduced from the classical Jacquet–Langlands correspondence. In fact, a version of Theorem A after localizing at a maximal ideal was first shown by Emerton [Reference EmertonEme14, 3.3.2] by reversing this argument: the classical correspondence gives rise to a surjective map of completed Hecke algebras

\[ {\mathbb{T}'}^\wedge_{M^{K^p}_{p\text{-}\mathrm{adic}}} \rightarrow {\mathbb{T}'}^\wedge_{\mathcal{A}^{K^p}_{\mathbb{Q}_p}} \]

(which is enough to obtain Corollary B), and then strong results in the deformation theory of Galois representations can be used to deduce that this map is an isomorphism after localizing at a maximal ideal $\mathfrak {m}$ (under minor hypotheses on $\mathfrak {m}$).

By contrast, our proof is based entirely on the $p$-adic geometry of modular curves. Thus, we obtain a new proof of Corollary B that is independent of the classical Jacquet–Langlands correspondence, and our proof of Theorem A does not use any $R=\mathbb {T}$ theorems or other results on Galois deformations.

1.3 Eigenspaces and the local $p$-adic Jacquet–Langlands correspondence

One failing of Theorem A as stated is that it says nothing about the $D^\times (\mathbb {Q}_p)$-representation appearing in the eigenspace in $\mathcal {A}^{K^p}_{\overline {\mathbb {Q}}_p}$ attached to a character of ${\mathbb {T}'}^\wedge _{\mathcal {A}^{K^p}_{\mathbb {Q}_p}}$ valued in a finite extension of $\mathbb {Q}_p$. Indeed, because completed Hecke algebras are formed by compiling congruences of eigensystems, which may not be reflected in congruences of eigenvectors, one does not even know whether such an eigenspace is non-empty. On the other hand, one expects that it is always non-empty, and that the $D^\times (\mathbb {Q}_p)$-representation appearing is essentially that constructed in the local correspondences of Knight [Reference KnightKni16] and Scholze [Reference ScholzeSch18].

In the course of the proof of Theorem A, we produce explicit eigenvectors that show this eigenspace is non-empty at least for eigensystems attached to classical modular forms. In the author's thesis [Reference HoweHow17], it was explained how to refine this construction so that it applies more generally to overconvergent modular forms, and it was also verified that the eigenvectors obtained are never locally algebraic for the action of $D^\times (\mathbb {Q}_p)$ (and, thus, in some sense are new, i.e. not obtainable by combining the classical Jacquet–Langlands correspondence and Gross's [Reference GrossGros99] theory of algebraic automorphic forms, which together furnish a complete description of the locally algebraic vectors). One can now do better in at least two ways.

  1. (i) An overconvergent modular form more canonically gives rise to a non-zero map to the corresponding eigenspace in quaternionic automorphic forms from a purely local representation of $D^\times (\mathbb {Q}_p)$ constructed as a space of distributions on the Lubin–Tate tower. The eigenvector referred to above is obtained as the image of a Dirac delta function under this map. By studying this local representation we can obtain considerably more, though still incomplete, information about the eigenspace.

  2. (ii) Combining this with recent results of Pan [Reference PanPan22] on the ubiquity of overconvergent modular forms, we find that, under some minor hypotheses on the associated Galois representation, the eigenspace is always non-empty.

Both points will be explained further in future work.

1.4 Outline

In § 2 we give some preliminaries, including in § 2.4 some results on comparing completed Hecke algebras that will be essential in the proof of Theorem A. In § 3 we set up our moduli problems for elliptic curves and recall the basic adelic setup for modular forms (classical, mod $p$, and $p$-adic). The main result of the section is Theorem 3.7.1, which identifies the supersingular Igusa variety with a quaternionic coset, one of the key ingredients in the proof of both the mod $p$ and $p$-adic correspondence. Some aspects of the way we set up our moduli problems may be of independent interest; for more, we refer the reader to the introduction of § 3.

In § 4 we prove the version of Serre's mod $p$ Jacquet–Langlands correspondence stated above as Theorem 1.1.1. The proof we give is essentially that of Serre, but we emphasize carefully from the beginning the role of uniformization of the supersingular locus by the supersingular Igusa variety, which, by the above, is just a quaternionic coset. In particular, modular forms can be evaluated to quaternionic automorphic forms after choosing a trivialization of the modular bundle along the Igusa variety. A mod $p$ modular form can have zeros or poles along the supersingular locus, but these can be cleared Hecke-equivariantly using the Hasse invariant in order to obtain a clean comparison of the corresponding Hecke algebras. When $p =2 \text { or } 3$, this proof actually falls just short of the full Theorem 1.1.1, but in § 4.5 we explain how to obtain the full statement as a consequence of Theorem A.

In § 5 we prove Theorem A. Here the quaternionic coset, again in its avatar as the supersingular Igusa variety, arises naturally as a fiber of the Hodge–Tate period map in the infinite level perfectoid modular curve (as in [Reference Caraiani and ScholzeCS17]). Thus, we can evaluate modular forms to $p$-adic quaternionic automorphic forms after choosing a trivialization of the modular bundle on this fiber. A simple argument shows this evaluation map is injective; the other key property we need is density of the image. We establish this with the help of Scholze's fake Hasse invariants.Footnote 2 These properties of the evaluation map are combined with the results of § 2.4 to deduce Theorem A.

2. Preliminaries

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’.

Remark 2.3.1 When $R$ is not normal, this is not quite the category of elliptic curves up-to-isogeny (respectively, prime-to-$p$ isogeny) considered in [Reference DeligneDel71, § 3], but rather the full subcategory consisting of objects with a genuine underlying elliptic curve. In general, one also formally enforces effectivity of étale descent. This full subcategory will suffice for our needs as our moduli problems typically include rigidifying data.

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.

Definition 2.4.1

  1. (i) An $L$-Banach space is a complete topological $L$-vector space $V$ whose topology is induced by an ultrametric norm; we refer to the choice of such a norm on $V$ as a Banach norm.

  2. (ii) A bounded collection of vectors $\{e_i \}_{i \in I}$ in an $L$-Banach space $V$ is an orthonormal basis if every $v \in V$ can be written uniquely as

    (2.4.1)\begin{equation} v = \sum_{i \in I} v_i e_i,\quad v_i \in L, v_i \rightarrow 0. \end{equation}
  3. (iii) We say that $V$ is orthonormalizable if it admits an orthonormal basis.

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].

Definition 2.4.2 For $V$ and $W$ two $L$-Banach spaces, we write $B(V,W)$ for the space of bounded (equivalently, continuous) linear operators from $V$ to $W$.

  1. (i) The choice of Banach norms on $V$ and $W$ induces an operator norm on $B(V,W)$ defined by

    \[ |T|=\sup_{v\in V, v\neq0} |T(v)|/|v|. \]
    The operator norms for different choices of Banach norms on $V$ and $W$ are equivalent and with the induced topology $B(V,W)$ is a Banach space.
  2. (ii) The topology of pointwise convergence Footnote 4 on $B(V,W)$ is defined by the family of seminorms $T \mapsto |T(v)|$ indexed by $v\in V$ (for any Banach norm on $W$).

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.

Definition 2.4.3 We say $\mathcal {T} \subset B(V,W)$ is bounded if it is bounded in the operator norm topology.

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 $w_i \in W_i$,

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

Lemma 2.4.6 Using notation as before, if the action of $A$ on each $W_i$ is bounded (i.e. the image of $A$ in $B(W_i, W_i)$ is bounded in the operator norm topology), then $\widehat {A}_{(W_i)_{i \in I}}$ is a closed subring of $\prod _{i \in I} B(W_i, W_i)$.

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

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

Proof. Immediate by applying Lemma 2.4.4 to an orthonormal basis and using the fact that an orthonormal basis remains an orthonormal basis under completed base change.

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

Lemma 2.4.9 Suppose $V$ is an orthonormalizable $L$-Banach space equipped with a bounded action of a ring $A$, and $(W_i)_{i \in I}$ is a collection of $A$-invariant closed subspaces such that the span of $\bigcup _i W_i$ is dense in $V$. Then $\widehat {A}_{(W_i)_{i\in I}} = \widehat {A}_{V}$.

Remark 2.4.10 In this setup, each $W_i$ is automatically a Banach space as a closed subspace of a Banach space and the action on $W_i$ is automatically bounded.

Proof. We abbreviate $\widehat {A}_W := \widehat {A}_{(W_i)_{i\in I}}\subset \prod B(W_i, W_i)$. We then obtain a map $\widehat {A}_V \rightarrow \widehat {A}_W$ via restriction: if the image of $(a_j)_{j \in J}$ in $B(V,V)$ converges to $T$, then, in particular, the image of $(a_j)_{j \in J}$ in $B(W_i, W_i)$ converges to $T|_{W_i}$. This restriction map is injective by the density hypothesis.

We show now that it is surjective. The key observation that makes this possible is that, by the density hypothesis, we may choose an orthonormal basis $(e_m)_{m \in M}$ for $V$ consisting of vectors $e_m$ each of which is a finite linear combination of vectors in the subspaces $W_i$: indeed, if we fix a pseudo-uniformizer $\pi$ in $\mathcal {O}_L$ and an arbitrary orthonormal basis $(f_m)_{m \in M}$, then any collection of vectors $(e_m)_{m \in M}$ with $|f_m - e_m| \leq |\pi |$ will also be an orthonormal basis.

Now, suppose $(T_i)_{i \in I} \in \widehat {A}_W$, and fix a net $a_j$ of elements of $A$ whose image converges to $(T_i)_{i \in I}$. Then we find that for each $m$, $\lim _j a_j \cdot e_m$ exists in $V$, call it $v_m$ and, by boundedness of the action, the set of $v_m$ is bounded. There is, thus, a unique bounded linear operator $T:V\rightarrow V$ such that $T(e_m)=v_m$. We conclude by Lemma 2.4.4 that the image of $a_j$ in $B(V,V)$ converges to $T$, and then by restriction that, in fact, $T|_{W_i}=T_i$.

Now, a topology is uniquely determined by the knowledge of which nets converge to which points. With this bijection established, Lemma 2.4.4 tells us that the same nets converge to the same points, so the bijection is a homeomorphism.

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.

Proof. It follows from Lemma 2.4.9 that $\widehat {A}_V = \widehat {A}_{(W_i)_{i \in I}}$. The result then follows from Lemma 2.4.7, because in this case the closure of the image of $A$ will be identified with the limit of the $A_{i,n}$ as a subset of the product appearing there.

3. Modular curves and Igusa varieties

In this section, we study some moduli problems for elliptic curves. In § 3.1 we give isogeny formulations for some classical moduli problems and recall the standard representability results. In § 3.2 we recall the construction of the modular bundle and the adelic representations on modular forms, as well as the construction of the Hasse invariant. In § 3.3, we recall the construction of the supersingular and ordinary loci on the mod $p$ modular curve. In § 3.4, we recall some Igusa moduli problems over the ordinary locus and their relation with mod $p$ and $p$-adic modular forms as developed by Katz [Reference KatzKat75a].

In §§ 3.53.7, we undertake a study of the supersingular Igusa variety, culminating with the identification of the supersingular Igusa variety with a quaternionic coset in Theorem 3.7.1. Everything here except this final identification is a very special case of results of Caraiani–Scholze [Reference Caraiani and ScholzeCS17]. However, following our treatment of modular curves, we take a resolutely ‘top-down’ approach, and for the most partFootnote 6 our treatment here is independent of the results of [Reference Caraiani and ScholzeCS17]. We lean instead on the Hasse invariant and other ideas specific to this special case.

As to the identification with a quaternionic coset, the basic idea is already present in Serre's letter [Reference SerreSer96], so our main contribution is a careful treatment by exploiting the group action at infinite level. This identification is a key ingredient in both the mod $p$ correspondence in § 4 and the $p$-adic correspondence in § 5.

Finally, we remark that, motivated by our specific needs, we have made what appear to be some non-standard choices in defining our moduli problems.

  1. (i) We allow level defined by an arbitrary closed adelic subgroup, which facilitates the free usage of large group actions on infinite-level moduli problems and, in particular, transparent passage between the infinite-level prime-to-$p$ moduli problem over $\mathbb {Z}_{(p)}$ and infinite-level moduli problem over $\mathbb {Q}$.

  2. (ii) We give an up to isogeny definition of level structure that does not require the base scheme to be locally noetherian (i.e. does not use the (pro)-étale fundamental group). In particular, this is necessary to allow arbitrary closed subgroups as before, but also allows us to evaluate on, e.g., perfectoid rings and other very non-noetherian objects without appealing behind the scenes to noetherian approximation.

We accomplish both of these goals by interpreting the sentence ‘level $K$ structure on $E$ is a $K$-orbit of trivializations of $V_{\mathbb {A}_f}(E)$’ literally, i.e. as the choice of a $K$-torsor in $\underline {\mathrm {Isom}}(\underline {\mathbb {A}_f}^2, V_{\mathbb {A}_f}(E))$. All representability statements are deduced from classical results on finite-level curves, and ultimately all of our arguments could be run in a more classical setup, as the diligent reader will have no trouble verifying.

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

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

Example 3.1.4 (Integral infinite level)

As in Example 3.1.2, when $K^p=\{e\}$, $\mathcal {K}^p$ is simply the choice of an isomorphism $\varphi _{\mathbb {A}_f^{(p)}}:\underline {\mathbb {A}_f^{(p)}}^2 \xrightarrow {\sim } V_{\mathbb {A}_f^{(p)}}(E)$. When $K^p=\{e\}$, we typically omit it from the notation and write simply $\mathfrak {Y}=\mathfrak {Y}_{\{e\}}$.

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

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

Example 3.1.6 Lemma 3.1.5 gives $\mathfrak {Y}_{\mathbb {Q}}=Y_{\mathrm {GL}_2(\mathbb {Z}_p)},$ where on the right-hand side $\mathrm {GL}_2(\mathbb {Z}_p)$ is viewed as a closed subgroup of $\mathrm {GL}_2(\mathbb {A}_f)$, and, as previously, $\mathfrak {Y}=\mathfrak {Y}_{\{e\}}$. This identification explains one reason why it is convenient to allow an arbitrary closed subgroup in the formulation of the moduli problem.

Definition 3.1.7 A closed subgroup $K \leq \mathrm {GL}_2(\mathbb {A}_f)$ (respectively, $K^p \leq \mathrm {GL}_2(\mathbb {A}_f^{(p)})$) is sufficiently small if it stabilizes a $\widehat {\mathbb {Z}}$-lattice $\mathcal {L} \subset \mathbb {A}_f^2$ (respectively, a $\widehat {\mathbb {Z}}^{(p)}$-lattice $\mathcal {L} \subset (\mathbb {A}_f^{(p)})^2$) and lies in the kernel of the map $\mathrm {GL}(\mathcal {L}) \rightarrow \mathrm {GL}(\mathcal {L}/n\mathcal {L})$ for some $n \geq 3$ (respectively, and $(n,p)=1$).

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, $K_1^p/K_2^p$).

Proof. We argue only in the case over $\mathbb {Q}$, as the argument over $\mathbb {Z}_{(p)}$ is essentially the same. If we fix a $\widehat {\mathbb {Z}}$ lattice $\mathcal {L}\subset \mathbb {A}_f^{2}$ preserved by $K$, then, as in [Reference DeligneDel71, Corollaire 3.5], we see that the moduli problem can be replaced with an equivalent up to isomorphism moduli problem by taking $\mathcal {K}$ in $\underline {\mathrm {Isom}}(\mathcal {L}, T_{\widehat {\mathbb {Z}}^{\bullet }}(E))$. The assertion that (3.1.1) is an isomorphism then amounts to the following: for $E/R$ an elliptic curve, if we consider the $\underline {\mathrm {GL}(\mathcal {L})}$-torsor $\mathcal {G}:=\underline {\mathrm {Isom}}(\mathcal {L}, T_{\widehat {\mathbb {Z}}^{\bullet }}(E))$, we must show that the following data are equivalent:

  1. (i) a $K$–torsor inside $\mathcal {G}$;

  2. (ii) a system of $K'$-torsors inside $\mathcal {G}$ for $K'$ compact open, $K \leq K' \leq \mathrm {GL}(\mathcal {L})$, compatible under inclusion.

This equivalence is provided by Lemma 2.2.1.

As we have established that (3.1.1) is an isomorphism, the rest of the claim for general $K$ is essentially formal if we can establish the representability claims for $K$ compact open. However, for $K$ compact open, we can conjugate to assume the lattice $\mathcal {L}$ as previously is $\widehat {\mathbb {Z}}^2$, and then the representability statements are consequences of the classical theory of finite level modular curves as in, e.g., [Reference Katz and MazurKM85].

Definition 3.1.9 (Compactified modular curves)

For $K \leq \mathrm {GL}_2(\mathbb {A}_f)$ (respectively, $K^p \leq \mathrm {GL}_2 (\mathbb {A}_f^{(p)})$) a sufficiently small compact open subgroup, we form compactifications $X_K$ (respectively, $\mathfrak {X}_{K^p}$) as in [Reference Katz and MazurKM85, 8.6] after fixing a lattice $\mathcal {L} \subset \mathbb {A}_f^2$ (respectively, $\mathcal {L} \subset (\mathbb {A}_f^{(p)})^2$) preserved by $K$ (respectively, $K^p$) to relate to classical finite-level moduli problems as in the previous proof. We obtain smooth projective curves $X_K/\mathbb {Q}$ (respectively, $\mathfrak {X}_{K^p}/\mathbb {Z}_{(p)}$), and the finite étale maps in the tower of $Y_K$ (respectively, $\mathfrak {Y}_{K^p}$) for $K$ (respectively, $K^p$) sufficiently small compact open extend to finite maps in the tower of $X_K$ (respectively, $\mathfrak {X}_{K^p}$).

It can be checked that the natural group actions also extend. For $X_K$ and $\mathfrak {X}_{K^p,\mathbb {F}_p}$ this is even immediate because the smooth compactifications of smooth curves are functorial over a perfect field: using this, we could also define $X_K$ and $\mathfrak {X}_{K^p,\mathbb {F}_p}$ with no reference to the moduli problem.

We extend these definitions to $K$ (respectively, $K^p$) sufficiently small closed by taking limits as in Proposition 3.1.8, and the resulting objects are schemes because the transition maps are affine.

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 $M_{k,\mathbb {F}_p}$.

Remark 3.2.1 Neither $M_{k,\mathbb {F}_p}$ nor the image of reduction is what is typically referred to as mod $p$ modular forms. We recall this definition in § 3.4.

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