Overconvergent modular forms and their explicit arithmetic

By Jan Vonk

Abstract

In these notes we aim to give a friendly introduction to the theory of overconvergent modular forms and some examples of recent arithmetic applications. The emphasis is on explicit examples and computations.

Introduction

The theory of -adic modular forms has its origins in the work of Serre and Katz in the 1970s, and has seen a spectacular amount of development and applications in number theory since then. In this note, we aim to provide its context and sketch the rudiments of the theory, adopting an approach where we favour explicit examples and computations over proofs. This is done with the hope that the uninitiated reader may build up some intuition and working knowledge as a stepping stone to the literature on the subject, which can be somewhat daunting to outsiders, but for which there is no substitute if one wants to become a serious user. We have included references to many of the original texts.

We should warn the reader that by its very design, this article is doomed to be incomplete, and several crucial developments are not discussed in this text. One notable example is that we have omitted a discussion of the theory of overconvergent modular symbols, which often provides an alternative framework that is in its own way highly suited for explicit computation. The author wishes to apologise for this omission, and many others, with the added clarification that this is merely a reflection of his own lack of experience with this approach. Likewise, many recent exciting developments in the area, such as the burgeoning topic of higher Hida theory, will unfortunately not be discussed in any detail here. Finally, it is important to note that a great many excellent (expository) sources on the theory of overconvergent modular forms already exist, which include for instance the beautiful treatments of Emerton Reference Eme11 and Calegari Reference Cal13.

1. Congruences between modular forms

We start by recalling some basic definitions and motivate the theme of these notes by discussing some classical congruences for the Ramanujan -function, a weight- modular form of level . These illustrate different types phenomena, and we highlight the features we wish to explore in these notes.

1.1. Modular forms

Suppose is a finite index subgroup. Then denotes the space of modular forms of weight , that is the space of holomorphic functions on the Poincaré upper half-plane which satisfy the transformation law

and are holomorphic at the cusps of . The subspace of cuspforms consists of those functions which vanish at all the cusps and is denoted by . In these notes, will usually be given by the congruence subgroup

Any modular form is invariant under translation and admits a Fourier expansion

This will be referred to as its -expansion, and the are called its Fourier coefficients. We refer to as the constant term of and the for as its higher Fourier coefficients. When , we say is normalised.

Classical examples of modular forms are given by the Eisenstein series, which are constructed as follows. For any even , we have the weight- normalised Eisenstein series

where is the th Bernoulli number (see equation Equation 29) and where is the divisor function. They define modular forms of weight for the full modular group . We note that for the series defining fails to converge absolutely, and indeed we have . We will see in §2.5 that the -expansion above still has meaning for as a -adic modular form.

The dimension of and may typically be calculated using the Riemann–Roch theorem, and the theory of modular symbols allows one to compute, to any desired -adic accuracy, a set of -expansions of a basis for it. For more details on these computations, see the detailed treatment of Stein Reference Ste07.

1.2. The Hecke algebra

Two central aspects of the theory of modular forms are the action of the Hecke algebra and their associated Galois representations, which we briefly discuss now.

The spaces of modular forms and are finite dimensional, and they are equipped with an action of the Hecke algebra, generated by operators for any prime , where it is customary to use the notation whenever . In terms of -expansions Equation 3 they are given by the expressions

Any modular form that is an eigenvector for all these Hecke operators is called an eigenform. The Eisenstein series defined in Equation 4 is a simple example of an eigenform, which satisfies

Note that the eigenvalue is equal to the divisor function , and therefore also the th Fourier coefficient of displayed in Equation 4. One can verify from the expressions Equation 5 that this is always the case; if is a normalised eigenform, then its eigenvalue for or is equal to its th Fourier coefficient. For an introduction to the basic properties of Hecke operators, see Diamond and Shurman Reference DS05.

Many spectacular results in number theory revolve around the notion of Galois representations. In what follows, this will always mean a continuous representation

where is the absolute Galois group of , and is either the field of complex numbers (in which case is called an Artin representation) or a -adic field such as . Important examples of the latter arise from elliptic curves. Suppose is an elliptic curve defined over , choose a prime , and consider the -adic Tate module, obtained from the inverse limit of the torsion points on of -power order,

which is a two-dimensional -vector space. This space has a natural action of the Galois group , given by the Galois action on the coordinates of the -torsion points on , which are algebraic numbers. Many important arithmetic properties of the elliptic curve can be recovered from this Galois representation. For instance, for any prime of good reduction for , we have that

In other words, the trace of the matrix of Frobenius at is related to the number of points of over the finite field . It is striking that this representation depends on the choice of a prime , but the traces of Frobenius at primes of good reduction for are integers and are independent of the choice of .

Suppose is an eigenform of level and weight , where is the conductor of . We say that is attached to if the traces of Frobenius elements Equation 9 are equal to the Fourier coefficients of . In other words, this means that for all but finitely many we have

Important developments, culminating in the work of Wiles Reference Wil95, Taylor and Wiles Reference TW95, and Breuil, Conrad, Diamond, and Taylor Reference BCDT01 show that for any elliptic curve over , there exists a modular form that is attached to it in this sense. This has led not just to a proof of Fermat’s last theorem, but subsequent developments continue to this day to settle long-standing conjectures in number theory.

Remark.

We briefly mention that the converse was known much earlier. That is, a construction of Eichler and Shimura attaches an elliptic curve to any eigenform of weight with integer Fourier coefficients. Generally, to any normalised cuspidal eigenform of weight one may attach a two-dimensional Galois representation of , which is unramified at all primes away from a finite set, and which satisfies

where is the th Fourier coefficient of . When this representation is valued in a nonarchimedean local field, and it was constructed by Deligne Reference Del71, though it is no longer attached to an elliptic curve as in the aforementioned construction for due to Eichler and Shimura. When , it is an Artin representation, constructed by Deligne and Serre Reference DS74 from the representations in higher weight via congruences.

1.3. Some examples of congruences

The Ramanujan -function is the unique normalised cusp form of weight for the group . Its -expansion is given by the infinite product due to Jacobi,

This explicit product allows us to easily establish a number of congruences between the Fourier coefficients of and those of various other modular forms, going back to the early twentieth century. For the reader who would like to inspect these manually, we tabulate its first few Fourier coefficients for prime:

Example 1.1.

We begin with a congruence that appears in the work of Ramanujan Reference Ram16. Consider the weight- Eisenstein series introduced in Equation 4. For , its constant term is equal to , whereas for the constant term is . Since the space is two dimensional, spanned by and , the form must be a linear combination of the two. Computing the first two terms of all three -expansions, we find that

and since all three modular forms involved have -integral -expansions, we obtain

In particular, we see that for any prime , we get the celebrated Ramanujan congruences

For a beautiful and very detailed expository discussion of this example in the broader context of ideal class groups of cyclotomic fields and Galois representations, see Mazur Reference Maz11.

Example 1.2.

This example is due to Wilton Reference Wil30, and it establishes a congruence modulo between and a certain form of weight . We have the following congruences⁠Footnote1 for ,

1

The reader familiar with the Dedekind -function—which is a modular form of weight for some character of the metaplectic double cover of —will recognise the form on the right-hand side as .

where the first is a consequence of the fact that for any prime , the binomial coefficient is divisible by for all , and the second follows from a calculation using the Euler identity

It is a classical result (see for instance Hecke Reference Hec26) that right-hand side of Equation 17 is a modular form of weight . It is in fact a Hecke eigenform, with an associated Artin representation that we can identify easily: the quadratic field has class number , and its Hilbert class field is obtained by adjoining a root of the cubic polynomial

which has discriminant . The natural quotient gives us

from the unique two-dimensional irreducible representation of . This is the two-dimensional Artin representation attached to the above weight- form. In particular, this means that the congruence class of modulo may be worked out from the splitting behaviour of the prime in the extension . The reader may enjoy verifying in general or simply checking on a few small values of in the table Equation 13 that this boils down to the statement⁠Footnote2 that for any prime , we have

2

We use the notation for the Legendre symbol for , which equals if is a square modulo , and otherwise.

Example 1.3.

As in the previous example, an elementary divisibility of binomial coefficients allows us to obtain from the infinite product expansion the following congruence for

The right-hand side is a weight- normalised cusp form of level . It is associated to the elliptic curve

so that we obtain in particular the following congruences for :

The reader may enjoy verifying this for a few small primes, using the table Equation 13 and equation Equation 23. Unfortunately, the law governing the association cannot be made explicit in the same elementary terms as in Equation 16 and Equation 21. The reason for this was explained by Shimura Reference Shi66, since this law is governed by the traces of the -adic representation attached to the elliptic curve Equation 23, and Shimura showed that its mod reduction has image . Since this group is not solvable, the law is equivalent to the splitting behaviour of primes in a nonsolvable extension of , which is in contrast with Example 1.2, where the relevant group was .

1.4. The context of this article

The three examples of congruences Equation 15, Equation 17, Equation 22 are of very different flavours, and they illustrate different but related phenomena that arise in the -adic theory of modular forms:

The first is a congruence between a cusp form and an Eisenstein series, of the same weight. Such congruences are central in Iwasawa theory, and related to the notion of the Eisenstein ideal; see Mazur Reference Maz77. We will not discuss this theme, but we mention that this is a fascinating topic that remains today an active area of research; see for instance Reference Mer96Reference CE05Reference Lec18Reference WWE20. A beautiful introduction to the ideas in this area can be found in Mazur Reference Maz11.

The second and third are both congruences between two cusp forms of different weights. This resonates with the framework of -adic families of modular forms, as developed by Reference Hid86bReference Hid86aReference Col97bReference CM98 and many others, and it is these types of congruences that form the focus of this document. We note that both these examples are of a very different nature. Example 1.2 exhibits a congruence between a modular form of weight and another of a higher weight,⁠Footnote3 and it results in an elementary description of the congruence class of modulo . Example 1.3 on the other hand exhibits a congruence with a form associated to an elliptic curve. There is no similarly elementary characterisation of the congruence class of the modulo .

3

The existence of such congruences is an important ingredient in the aforementioned work of Deligne and Serre Reference DS74 on the existence of Artin representations attached to modular forms of weight .

In these notes, we will focus primarily on the theme of congruences between modular forms of different weights and -adic families. Traditionally, the theory was built around the prototypical example of the Eisenstein family, as in Coleman Reference Col97b, until more recent advances due to Pilloni Reference Pil13 and Andreatta, Iovita, and Stevens Reference AIS14 on the geometric interpolation of line bundles, which allows us to develop the theory abstractly, without relying on the Eisenstein family. From a practical and computational point of view, this family remains of primordial importance, so the next section will quickly review it, motivated by the strategy of Serre to show the existence of the Kubota–Leopoldt -adic L-function.

2. Kummer congruences and Eisenstein series

We begin with a brief discussion of the Kummer congruences, and introduce Serre’s important idea of inferring the -adic variation of the constant term of a modular form, from that of its higher Fourier coefficients. This idea appeared in Serre Reference Ser73 and goes back to observations of Hecke Reference Hec24 and Siegel and Klingen Reference Kli62Reference Sie68. It will make several appearances throughout these notes.

2.1. The Kummer congruences

Recall that the Riemann zeta function may be analytically continued to the entire complex plane, except for a simple pole with residue at the point . It satisfies the functional equation

Of special importance are its values at negative odd integers (or equivalently, by the functional equation, at positive even integers), which were computed first by Euler in 1734 and read on 5 December 1735 in the St. Petersburg Academy of Sciences. The starting point for Euler was the easily verified identity

By taking the logarithmic derivative, we obtain the identities

On the other hand, the Bernoulli numbers are defined via the generating series

and hence we can formally extract the even part of this series as

Bearing in mind that , we obtain the identity

It now follows formally from Equation 28 and Equation 32 that

and hence by the functional equation

The fact that the value of the zeta function at negative odd integers is a rational number is remarkable. We will revisit this in the more general setting of L-functions of totally real number fields, when we discuss the explicit formula for this rational number obtained by Klingen and Siegel Reference Kli62Reference Sie68, following an idea of Hecke Reference Hec24 using diagonal restrictions of Hilbert Eisenstein series. This is discussed in §4.5.

The Bernoulli numbers have interesting -adic properties, notably by two results established in the mid-nineteenth century which are the starting point for our investigations: the Clausen–von Staudt theorem Reference Cla40Reference vS40 and the Kummer congruences Reference Kum51. For convenience, we assume henceforth that .

Lemma 2.1.

If are two positive even integers such that , then