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.
The theory of 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. -adic
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 a -function,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 , and the -expansion 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 Bernoulli number (see equation thEquation 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 above still has meaning for -expansion as a modular form. -adic
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 accuracy, a set of -adic of a basis for it. For more details on these computations, see the detailed treatment of Stein -expansionsReference 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 . -expansionsEquation 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 , Fourier coefficient of th 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 Fourier coefficient. For an introduction to the basic properties of Hecke operators, see Diamond and Shurman thReference 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 field such as -adic Important examples of the latter arise from elliptic curves. Suppose . is an elliptic curve defined over choose a prime , and consider the , Tate module, obtained from the inverse limit of the torsion points on -adic of order, -power
which is a two-dimensional space. This space has a natural action of the Galois group -vector given by the Galois action on the coordinates of the , points on -torsion 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. ,
1.3. Some examples of congruences
The Ramanujan is the unique normalised cusp form of weight -function for the group Its . is given by the infinite product due to Jacobi, -expansion
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:
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 theory of modular forms: -adic
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 families of modular forms, as developed by -adicReference 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 .
In these notes, we will focus primarily on the theme of congruences between modular forms of different weights and families. Traditionally, the theory was built around the prototypical example of the Eisenstein family, as in Coleman -adicReference 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 L-function. -adic
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 variation of the constant term of a modular form, from that of its higher Fourier coefficients. This idea appeared in Serre -adicReference 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 ,
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 properties, notably by two results established in the mid-nineteenth century which are the starting point for our investigations: the Clausen–von Staudt theorem -adicReference Cla40Reference vS40 and the Kummer congruences Reference Kum51. For convenience, we assume henceforth that .