A unipotent circle action on -adic modular forms

By Sean Howe

Abstract

Following a suggestion of Peter Scholze, we construct an action of on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the -adic modular curve whose ring of functions is Serre’s space of -adic modular functions. This action is a local, -adic analog of a global, archimedean action of the circle group on the lattice-unstable locus of the modular curve over . To construct the -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates ; along the way we also prove a natural generalization of Dwork’s equation for extensions of by valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of integrates the differential operator coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and -adic -functions.

1. Introduction and analogy

In this work, following a suggestion of Peter Scholze, we descend the unipotent quasi-isogeny action on a component Caraiani-Scholze’s Reference 1, Section 4 ordinary (big) Igusa formal scheme for to construct an action of the formal -adic torus on the Katz moduli problem over the ordinary locus. Suitably interpreted, this action is a local, -adic analog of the global, archimedean phenomena whereby the horizontal translation action of on the complex upper half plane descends to an action of on the image of in the complex modular curve.

The space of functions on the Katz moduli problem that are holomorphic at the cusps is equal to the completion of classical modular forms for the -expansion topology (Serre’s space of -adic modular functions⁠Footnote1 In the body of this text, we reserve the term -adic modular forms for those -adic modular functions with a weight, i.e. that transform via a character under the -action.). Thus we may think of our -action as a unipotent circle action on -adic modular functions. The analogy with the archimedean circle action is stronger than one might first guess, and leads, e.g., to interesting representation-theoretic consequences.

After constructing the -action, we study its properties and interaction with other classical notions in the -adic theory of modular curves and modular forms such as the unit root splitting, Dwork’s equation , the differential operator , Gouvea’s twisting measure, and Katz’s Eisenstein measures.

We highlight one application to explain the significance of this construction: via -adic Fourier theory, the -action is equivalent to the -adic interpolation of powers of polynomials in the differential operator . This allows us to introduce a twisting direction into any -adic family of modular forms. In particular, when applied to Eisenstein series, it allows the construction of certain two-variable -adic -functions studied by Katz Reference 9 starting from single-variable Eisenstein families. A key advantage of this method is that we construct the -action and then relate it to differential operators obtained from the Gauss-Manin connection without ever using a cuspidal or Serre-Tate ordinary local expansion. In particular, our method will generalize to the -adic interpolation of certain differential operators constructed by Eischen and Mantovan Reference 4 on the -ordinary locus of more general PEL Shimura varieties (where local expansions are unavailable or difficult to work with) into actions of Lubin-Tate formal groups.

In the present work we have focused on exploring the ramifications of the existence of a large quasi-isogeny action on the Caraiani-Scholze Igusa formal scheme for the classical space of Katz/Serre -adic modular functions. In a sequel Reference 6, we study the action of the quasi-isogeny group on the space of functions on the big Igusa formal scheme itself as a natural space of -adic automorphic forms in the context of the -adic Langlands program. Ordinary -adic modular forms (in the sense of Hida) play an important role in this study, and in Reference 6 we also explain how Hida’s finiteness and classicality theorems for ordinary -adic modular forms can be understood from this perspective.

1.1. An archimedean circle action

Before stating our results, we explain the analogous archimedean circle action more carefully; this will help to motivate and clarify the -adic constructions that follow. Consider the complex manifold

Two important observations about follow immediately from Equation 1.1.0.1:

(1)

Modular forms of level (for any ) restrict to -invariant functions , and thus induce holomorphic functions on .

(2)

The action of by horizontal translation on descends to a (real analytic) action of the circle group on . This action integrates the vector field .

We can decompose any holomorphic function on according to this -action uniquely as a Fourier series

In other words, the space of functions on is a Fréchet completion of the direct sum of the character spaces for the -action, with each character appearing exactly once.

1.1.1. Fourier coefficients and representation theory

The Fourier coefficients of classical modular forms play an important role in the global automorphic representation theory for . In particular, for a Hecke eigenform, the constant coefficient is non-vanishing if and only if the corresponding global automorphic representation is globally induced (i.e. the modular form is Eisenstein). Suitably interpreted, the constant term is a functional that realizes the induction. The non-constant coefficients, on the other hand, are Whittaker functionals.

1.1.2. The slope formalism on metrized tori

While the construction of above may at first seem ad hoc, it has a natural moduli interpretation, which we explain now. The key point is to use the slope formalism for metrized tori, or, equivalently, lattices, as explained, e.g., in Casselman’s survey Reference 2.

A metrized torus is a finite-dimensional torus (compact real abelian Lie group) together with a translation invariant metric, or, equivalently, a positive definite inner product on . There is a natural slope formalism on metrized tori: the rank function is dimension, and the degree function is given by

If a two-dimensional metrized torus is unstable (i.e. not semistable), then it contains a unique circle of shortest length.

If is an elliptic curve, the underlying real manifold of is a two-dimensional metrized torus when equipped with the metric coming from the canonical principal polarization.

Example 1.1.3.

Consider the usual fundamental domain

for the action of on . For , let

We compute the values of for which is semistable: the metric induced by the principal polarization is identified with times the metric induced by the identity

and the standard metric on . Semistability is preserved by scaling the metric, so we may eliminate the scaling and consider just the metric induced by the standard metric on . The length of a shortest circle in is equal to the length of a shortest vector in , which is . The area of the entire torus , on the other hand, is . Thus, the slope of the full torus is , while the smallest slope of a circle inside is . We conclude that for , is semistable when , and otherwise is unstable with shortest circle given by

1.1.4. Moduli of unstable elliptic curves

Using the slope formalism, we may consider the moduli space of unstable elliptic curves equipped with a trivialization of the shortest circle, . From Example 1.1.3, we find that this space is naturally identified with . In this moduli interpretation, the space is the cover where the trivialization of the shortest circle is extended to an oriented trivialization . From the moduli perspective, the fact that we can evaluate modular forms to obtain functions on comes from two facts:

(1)

Given a point of , there is a unique holomorphic differential whose pullback to along the trivialization of the shortest circle integrates to . Thus, the modular sheaf is canonically trivialized over , and modular forms can be evaluated along this trivialization.

(2)

Using the polarization, the trivialization of the shortest circle also gives rise to a trivialization of the quotient torus , so that is equipped with the structure of an extension of real tori

The basis for the torsion on then gives rise to a canonical -level structure on for any level .

1.1.5. de Rham cohomology

Consider the extension structure Equation 1.1.4.1 on the universal elliptic curve over . The global section of the de Rham cohomology of is flat, so we obtain via pullback of a canonical flat section over . Moreover, because the image of in under pullback is (by definition) , which is flat, we find that is in the span of . Thus, is a holomorphic differential form on .

For the elliptic curve as in Example 1.1.3, if we denote by and the natural basis elements for and by and the dual basis, we find that , and , so that

In particular, the -action integrates the vector field dual to .

1.2. Statement of results

In this section we state our main results.

1.2.1. Dictionary

As we introduce the objects appearing in the local, -adic theory, it may be helpful to keep in mind the following dictionary for our analogy with the global, archimedean story:

Global, archimedean Local, -adic
as a metrized torus The -divisible group
Unstable two-dimensionalmetrized torus Ordinary height two -divisible group
The shortest circle in The formal group
Trivialization of the shortest circle Trivialization of the formal group
, , , ,
Canonical level structure Canonical arithmetic level structure
The Katz formal scheme
The (polarized) Igusa formal scheme of Caraiani-Scholze
Action of on Action of on
Action of on Action of the universal cover of on
Fourier series Sheaf over
Constant term Fiber at 0
Eisenstein series Ordinary -adic modular form

1.2.2. The Katz moduli problem and -adic modular forms

For a -adically complete ring, let be the category of -algebras in which is nilpotent. We consider the Katz moduli problem on classifying, over , triples

where is an elliptic curve up to prime-to- isogeny, is a trivialization of the formal group of ,

and is a trivialization of the adelic prime-to- Tate module.

By work of Katz Reference 8, the moduli problem is represented by a -adic formal scheme

where is a -adically complete flat (torsion-free) -algebra. For a -adically complete -algebra, we write

so that

There is a natural moduli action of on , where acts by composition with , and acts by composition with . For a continuous character of with values in , the eigenspace is a natural space of -adic modular forms of weight .

In particular, classical modular forms of integral weight and prime-to- level over (interpreted using sections of the standard integral model of the modular sheaf and curve) are embedded -equivariantly (up to a twist) in this space for the character . Concretely, a classical modular form of prime-to- level and weight gives rise to an element of by evaluating on the triple

Here is the universal triple parameterized by the identity map on and is an invariant differential form on . In other words, the modular sheaf on is trivialized by the canonical section

which allows us to evaluate classical modular forms after pullback to .

1.2.3. de Rham cohomology

We write

for the universal elliptic curve up-to-prime-to--isogeny. We have the relative de Rham cohomology

equipped with Hodge filtration

and Gauss-Manin connection .

Note that the moduli problem classified by is equivalent to the moduli problem classifying triples where and are as before, and is a trivialization of the prime-to- Tate module

all considered up to isomorphism of . Using this equivalence, we obtain a well-defined Weil pairing on , and combining this with the trivialization , we obtain the structure of an extension

This is analogous to the archimedean extension Equation 1.1.4.1. In particular, we obtain an extension of Dieudonné crystals

which we view as an extension of vector bundles with connection on . The sub-bundle is the unit root filtration, and it has a canonical basis element . If we identify with via the crystalline-de Rham comparison, then the Hodge filtration splits the extension of vector bundles and the image of in is flat. So, and give a basis for such that is lower nilpotent, and thus determined by a single differential form

By the theory of Kodaira-Spencer, the differential form is non-vanishing, and thus admits a dual vector field such that .

1.2.4. The -action.

Our main result, Theorem A below, shows that the vector field can be integrated to an action of on , and explains how this action interacts with the action of To state it, we will need the unramified determinant character defined by

Theorem A.

There is an action of on whose derivative is the vector field defined above. Moreover, this action combines with the action of to give an action of

where the semidirect product is formed with the respect to the conjugation action

Remark 1.2.5.

The -action of Theorem A is uniquely determined by the condition that it integrates . We note that acts as the derivation on cuspidal -expansions; however, in our proof we construct the action and prove it integrates without using cuspidal (or Serre-Tate) -expansions, which is an important point for future generalizations.

The key observation in the construction of this -action and subsequent computations is that we may work on a very ramified cover, a component of the (big) Igusa formal scheme of Caraiani-Scholze Reference 1, Section 4, where the extension structure Equation 1.2.3.1 extends to a trivialization of the -divisible group

At the price of the ramification, life is simplified on this cover: for example, computations with the crystalline connection are reduced to computing the crystalline realization of maps . Most importantly, the obvious action of automorphisms of on this cover extends to an action of a much larger group of quasi-isogenies.

This quasi-isogeny group contains a very large unipotent subgroup, the quasi-isogenies from to , or, the universal cover in the language of Scholze-Weinstein Reference 16. The action of this large group of quasi-isogenies is the ultimate source of the -action on . Indeed: is the quotient by the subgroup of isomorphisms, i.e. the Tate module , and thus picks up a residual action of

Remark 1.2.6.

The action of a larger group of quasi-isogenies on this cover is a natural characteristic analog of the prime-to-characteristic phenomenon where, when full level is added at , there is an isogeny moduli interpretation that gives an action of extending the action of in the isomorphism moduli interpretation. Rigidifying in characteristic using isomorphisms to an ordinary -divisible group provides both more and less structure than when : on the one hand, the isogeny group is solvable, and thus appears more like the subgroup of upper triangular matrices, but on the other hand the unipotent subgroup has a much richer structure than any groups that appear when . If we instead rigidified using a height two formal group, we would obtain a super-singular Igusa variety, which has more in common with the case (the isogeny action is by the invertible elements of the non-split quaternion algebra over ); in Reference 7 we use this structure to compare -adic modular forms and continuous -adic automorphic forms on the quaternion algebra ramified at and .

Remark 1.2.7.

In this remark we explain a connection to perfectoid modular curves. The generic fiber of the big Igusa formal scheme is a twist of a component of the perfectoid ordinary locus over . This component admits a natural action of the group of upper triangular matrices

which is identified over with an action of on the generic fiber of the big Igusa formal scheme.

Using this, the action of the -power roots of unity , an infinite discrete set inside of the open ball , on functions on the generic fiber of can be identified with the action of the natural Hecke operators on the invariants under

in functions on this component of the perfectoid ordinary locus. Thus, the -action extends the obvious action of to an action of a much larger group. We will not use this connection to perfectoid modular curves in the present work, however, it will play an important role in Reference 6.

1.2.8. Local expansions

An important aspect of our proof of Theorem A is that we make no appeal to local expansions at cusps or ordinary points, so that our approach is well-suited for generalization to other PEL Igusa varieties. After proving Theorem A, however, we also give a direct computation of the action on local expansions: we find that at ordinary points the action is given by multiplication of a Serre-Tate coordinate, and at the cusps it is given by multiplication of the inverse of the standard cuspidal coordinate .

1.2.9. Dwork’s equation

While developing some of the machinery used to compute the local expansions of the -action, and using the same philosophy of base change to a very ramified cover, we also give a new proof of Dwork’s equation on the formal deformation space of over which is valid for a larger family of Kummer -divisible groups (which include not only the deformations of over Artinian -algebras, but also, e.g., the -divisible group of the Tate curve, and other interesting groups when the base is not Artinian). These results can be found in Section 3.

1.2.10. Other constructions

This action can be constructed in at least three other ways, two of which have been discussed previously in the literature:

(1)

After preparing an earlier version of this article, we learned that Gouvea Reference 5, III.6.2 had already some time ago constructed a twisting measure equivalent to our -action (interpreted as an algebra action via -adic Fourier theory as described in 1.2.11 below). In 7.2 we recall Gouvea’s construction and explain how it can be rephrased as an alternate construction of the -action via the exotic isomorphisms of Katz Reference 9, 5.6. Gouvea’s construction has the advantage of using only classical ideas, but is conceptually more opaque. In particular, we note that the interaction of the -action with the prime-to- group action (equivalently, Hecke operators away from ) is considerably clarified by our construction.

(2)

While the current version of this article was under review, we learned that our -action is also a special case of a result of Liu-Zhang-Zhang Reference 13, Proposition 2.3.5, who gave a construction of a Lubin-Tate action on more general Shimura curves by using the Baer sum of extensions. There are some issues with the proof as written in loc. cit. because of a mistake in the statement of Serre-Tate theory Reference 13, Theorem B.1.1 over rings where is nilpotent (where one must allow for unipotent quasi-isogenies as well as isomorphisms, consider e.g. the base ). The connection between these two constructions will be elaborated further in future work of the author constructing actions on -ordinary Igusa varieties.

(3)

The simplest and most opaque approach is to build the -action algebraically starting with the differential operator and the -expansion principle; we explain this in Remark 1.2.12 below.

1.2.11. The algebra action

Via -adic Fourier theory, the action of described in Theorem A is equivalent to an action of on . This action admits a particularly simple description on cuspidal -expansions: acts as multiplication by on the coefficient of (cf. Theorem 7.1.1). As remarked above, the existence of this algebra action was first established by Gouvea Reference 5, Corollary III.6.8, who interpreted it as a twisting measure.

From this perspective, the action of the monomial function is by the derivation (recall ), and thus we may view our -action as interpolating the differential operators into an algebra action. In Section 8 we adopt this perspective to reinterpret some results of Katz Reference 9 on two-variable Eisenstein measures.

Remark 1.2.12.

In fact, we can construct the -action by applying the -expansion principle Reference 9, 5.2 to complete the action of polynomials in on to an action of . Note that polynomials are not dense , so the -expansion principle needed here says not just that the -expansion map is injective, but also that the cokernel is flat over .

In order to use this method, one must first show that the operator on -expansions preserves the space of -adic modular forms (instead of deducing this by differentiating the -action). One way this can be done is by showing it is the effect on -expansions of the differential operator dual to the image of under the Kodaira-Spencer isomorphism, which can be verified by a computation over , as explained by Katz Reference 9, 5.8.

1.2.13. Ordinary -adic modular forms

The action of interacts naturally with the -action on , and thus we may view as a -equivariant quasi-coherent sheaf on the profinite set (viewed as a formal scheme whose ring of functions is ). As is the space of characters of , this viewpoint is analogous to thinking of functions on in the global, archimedean setting as Fourier series.

A straightforward computation with -expansions implies that restriction induces an isomorphism between the fiber at of the subsheaf of -adic modular functions with -expansion holomorphic at all cusps and the space of ordinary -adic modular forms à la Hida. Note that the fiber at zero is the maximal trivial quotient for the -action, and ordinary modular forms are those such that the corresponding -adic Banach representation of admits a map to a unitary principal series. Thus, our statement is a local, -adic analog of the global, archimedean statement that the global automorphic representation attached to a classical modular form is globally induced if and only if its Fourier expansion has a non-zero constant term.

We do not discuss this phenomenon further in the present work, but this characterization of ordinary -adic modular forms will play an important role in our study of functions on as a natural space of -adic automorphic forms in Reference 6. Moreover, this perspective also leads to representation-theoretic proofs of Hida’s finiteness and classicality results for ordinary -adic modular forms, as will be explained in Reference 6.

1.3. A remark on notation

Over a ring in which is topologically nilpotent, the formal group is equivalent to the -divisible group