I – Motivation and overview
The subject of modular forms is vast and has applications to number theory, algebraic geometry, combinatorics, and many other areas of mathematics and physics.
Modular forms are traditionally viewed as holomorphic functions defined on the upper half plane with remarkable transformation properties. You may wonder how these holomorphic functions have anything to do with number theory. One interesting connection is that certain cubic equations called elliptic curves arise from modular forms.
An elliptic curve can be viewed as a plane algebraic curve over a field defined by an equation of the form : . Usually is chosen to be an algebraically closed field, such as the algebraic closure of or . If we let , then the work required for the proof of Fermat’s Last Theorem by Wiles showed that a certain class of elliptic curves over , called semi-stable elliptic curves, arise from modular forms. This program was carried on to prove the full modularity theorem by Breuil, Conrad, Diamond, and Taylor who showed that all elliptic curves over are modular so they arise from modular forms.
The picture below is the elliptic curve plotted in Sage.
There are a few lenses through which one can view the phrase “arise from modular forms.” One of the most concrete ways is to reinterpret the equation as coming from a differential equation satisfied by the so-called Weierstrass -function, . Using the meromorphic function, , one can show and the coefficients and can be viewed as values of Eisenstein series which are important examples of modular forms. Therefore, the coefficients are values of modular forms.
A second way of viewing this statement is via -functions. An -function is a complex-valued function called a Dirichlet series of the form
Number Theorists have observed that most -functions have many interesting attributes, such as an Euler product, a functional equation, and an analytic continuation of the domain of definition to the whole complex plane. The most famous example of an -function is the Riemann Zeta Function . The phrase “elliptic curves over arise from modular forms or are modular” means that the -function coming from an elliptic curve and a corresponding a modular form are the same. Therefore, .
A third way to view this statement is through Galois Representations. These are representations where is usually a number field. In general, a Galois representation is a representation of any Galois group. Similar to -functions, the Galois Representations arising from modular forms are isomorphic as modules to those from elliptic curves, so that if is a prime satisfying some technical conditions, then .
II – The Modular Group and its Transformations
Before we jump into the details of modular forms, let us mention the famous quote of Barry Mazur.
Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist. – Barry Mazur
Let us now explore these surprisingly symmetric functions. Modular forms rely on the transformations of the familiar group
called the Modular Group. These matrices in can be viewed as automorphisms of the Riemann sphere by the fractional linear transformation on a complex variable defined as
Then if and approaches then we can think of mapping to since as then the denominator approaches so the fraction grows rapidly to . Furthermore, taking the limit of shows that maps to in the traditional sense of a limit. Also, if then . Now that we have this transformation, we can plug in different matrices in to reveal its symmetries. Let denote a matrix in .
You can check that if or , then we obtain , as expected. Actually, in general, if you have two matrices and , they will give the same transformation . The important symmetries of this transformation are revealed when we focus on the generators of the modular group. The standard generators of are the matrices
Now let’s plug these matrices into the transformation. We obtain
giving rise to a translation action and an inversion. These symmetries are very interesting and turn out to be a central feature of modular forms. Above we considered a complex variable . However, modular forms have the upper half plane as their domain, where .
III – Weak Modularity and Modular Forms
We will first define a weakly modular function. Before doing this let’s review some of complex analysis. A function defined on is called holomorphic if for every point there is an open neighborhood of where is complex differentiable. Another important type of function is a meromorphic function which is slightly weaker than a holomorphic function. A function defined on is called meromorphic if for every is complex differentiable at all but finitely points in every open neighborhood of .
Now let’s define a weakly modular function.
Definition: Let and let be a meromorphic function. Then we say is a weakly modular function of weight k if
where and .
The term is called the factor of automorphy which is assumed to be nonzero. This factor essentially measures how far the function is from invariance. This depends on the weight so if is large then varies from being invariant by a larger factor. Given this definition we can now generalize our earlier findings that and under the two generators of
I claim that if is a weakly modular function of weight then we have the following identities:
These identities follow from plugging in the two generators and of into the weakly modular definition. First, we have . The second identity is given by .
If , we obtain invariance and the case when is used in the proof of the modularity theorem. The first identity shows that modular forms are -periodic functions. We will soon see that classical modular forms are weakly modular forms with a bit more structure. The main difference between weakly modular forms and modular forms is that modular forms are required to also be holomorphic at . These modular forms are called holomorphic or classical modular forms. However, there are functions that are neither holomorphic nor weakly modular but exhibit symmetries similar to those of classical modular forms. The large umbrella term for classical modular forms and these other functions is automorphic forms. Let us investigate what holomorphic at means.
There are two standard approaches for understanding the holomorphic at condition. The first approach is the local approach. Let . First, change coordinates from to where such that is in the punctured
disk and define the function . Then is holomorphic on the punctured disk since and are holomorphic there. Then we can view the point at as the origin of the punctured disk . Then is holomorphic at means that the function extends holomorphically to the origin. The second interpretation looks at the Fourier expansion of a modular form and this approach is more global. We will impose a growth condition on . As before, if , then so if and only if . Now, since is holomorphic on the complex punctured unit disk, then it possesses a Fourier expansion . Then since is equivalent to , we don’t have to compute the Fourier expansion of . Instead, we must show that exists or that is bounded as . This discussion leads us to the definition of a holomorphic modular form.
Definition: Let . Then a function is a modular form of weight if
(I) is holomorphic on
(II) is weakly modular of weight
(III) is holomorphic at
Now that we have defined modular forms of weight , it is natural to ask what structure the space of modular forms possesses. The set of modular forms of a fixed weight , , forms a vector space over with the usual function addition. Two important properties of each vector space are that each vector space is finite-dimensional because of the holomorphic at condition, this requires some work, and that given two modular forms and of weights and , respectively, the product has weight . Also, there are no modular forms of negative weight so we only have to look at weight zero and above. Then to get the vector space of all modular forms of each weight , we take the direct sum of these complex vector spaces to obtain
which is a graded ring.
In the next post I will give explicit examples of modular forms and discuss congruence subgroups.
 D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 1998
 F. Diamond and J. Shurman, A First Course in Modular Forms, Springer GTM, 2005
 D. Goldfeld, Automorphic forms and L-functions for the group GL (n, R), Cambridge University Press, 2006
 D. Zagier, The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway, Springer, 2008