May | 2020 | Theories and Theorems

We have previously mentioned modular forms in The Moduli Space of Elliptic Curves and discussed them very briefly in the context of modular curves in Shimura Varieties. In this post, we will discuss this very important and central concept in modern number theory in more detail.

First we recall some facts about the group $\text{SL}_{2}(\mathbb{Z})$ , which is so important that it is given the special name of the modular group. It is defined as the group of $2\times 2$ matrices with integer coefficients and determinant equal to $1$ , and it acts on the upper half-plane (the set of complex numbers with positive imaginary part) in the following manner. Suppose an element $\gamma$ of $\text{SL}_{2}(\mathbb{Z})$ is written in the form $\left(\begin{array}{cc}a&b\\ c&d\end{array}\right)$ . Then for $\tau$ an element of the upper half-plane we write

$\displaystyle \gamma(\tau)=\frac{a\tau+b}{c\tau+d}$

A modular form (with respect to $\text{SL}_{2}(\mathbb{Z}))$ is a holomorphic function on the upper half-plane such that

$\displaystyle f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)$

for some $k$ and such that $f(\tau)$ is bounded as the imaginary part of $\tau$ goes to infinity. The number $k$ is called the weight of the modular form. If the function is not required to be bounded as the imaginary part of $\tau$ goes to infinity it is a weakly modular form, and if furthermore it is merely required to be meromorphic, , it is a meromorphic modular form. A meromorphic modular form of weight $0$ is just a meromorphic function on the upper half-plane which is invariant under the action of $\text{SL}_{2}(\mathbb{Z})$ (and bounded as the imaginary part of its argument goes to infinity) – we also call it a modular function.

We denote the set of modular forms of weight $k$ with respect to $\text{SL}_{2}(\mathbb{Z})$ by $\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}))$ . Adding together two modular forms of the same weight gives another modular form of the same weight, and modular forms can be scaled by a complex number, so $\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}))$ actually forms a vector space. We can also multiple a modular form of weight $k$ with a modular form of weight $l$ to get a modular form of weight $k+l$ , so modular forms of a certain weight form a graded piece of a graded ring $\mathcal{M}(\text{SL}_{2}(\mathbb{Z})$ :

$\displaystyle \mathcal{M}(\text{SL}_{2}(\mathbb{Z}))=\bigoplus_{k}\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}))$

Modular functions are actually functions on the moduli space of elliptic curves – but what about modular forms of higher weight? It turns out that he modular forms of weight $2$ correspond to coefficients of differential forms on this space. To see this, consider $d\tau$ and how the group $\text{SL}(\mathbb{Z})$ acts on it:

$\displaystyle d\gamma(\tau)=\gamma'(\tau)d\tau=(c\tau+d)^{-2}d\tau$

where $\gamma'(\tau)$ is just the usual derivative of he action of $\gamma$ as describe earlier. For a general differential form given by $f(\tau)d\tau$ to be invariant under the action of $\text{SL}(\mathbb{Z})$ we must therefore have

$\displaystyle f(\gamma(\tau))=(c\tau+d)^{2}f(\tau)$ .

The modular forms of weight greater than $2$ arise when we consider products of these differential forms. More technically, modular forms are sections of line bundles on modular curves, which come about when we compactify moduli spaces of elliptic curves (possibly with extra structure).

Let us now look at some examples of modular forms. Since modular forms “live on” moduli spaces of elliptic curves, we will keep in mind elliptic curves as we look at these examples. Our first family of examples are Eisenstein series of weight $k$ , denoted by $G_{k}(\tau)$ which is of the form

$\displaystyle G_{k}(\tau)=\sum_{(m,n)\in\mathbb{Z}^{2}\setminus (0,0)}\frac{1}{(m+n\tau)^{k}}$

Any modular form can in fact be written in terms of Eisenstein series $G_{4}(\tau)$ and $G_{6}(\tau)$ .

Now, let us relate this to elliptic curves. An elliptic curve over the complex numbers may be written as a Weierstrass equation

$\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}$

The coefficients on the right-hand side $g_{2}$ and $g_{3}$ are in fact modular forms, of weight $4$ and weight $6$ respectively, given in terms of the Eisenstein series by $g_{2}(\tau)=60G_{4}(\tau)$ and $g_{3}(\tau)=140G_{6}(\tau)$ .

Another example of a modular form is the modular discriminant of an elliptic curve, as a modular form denoted $\Delta(\tau)$ . It is a modular form of weight $12$ , and can be expressed via the elliptic curve coefficients that we defined earlier:

$\Delta(\tau)=(g_{2}(\tau))^{3}-27(g_{3}(\tau))^{2}$ .

Our final example in this post is not of a modular form, but a meromorphic modular form of weight $0$ , i.e. a modular function. It is holomorphic on the upper half-plane, but goes to infinity as the imaginary part of $\tau$ goes to infinity. It is the j-invariant associated to an elliptic curve. Once again we may express it in terms of the elliptic curve coefficients $g_{2}$ and $g_{3}$ :

$\displaystyle j(\tau)=1728\frac{(g_{2}(\tau))^{3}}{(g_{2}(\tau))^{3}-27(g_{3}(\tau))^{2}}$

Note that the denominator is also the modular discriminant. The points of the moduli space of elliptic curves correspond to isomorphism classes of elliptic curves, and since the j-invariant is an honest-to-goodness holomorphic function on the moduli space of elliptic curves over $\mathbb{C}$ , we can see that isomorphic elliptic curves will have the same j-invariant. This is not the case for the other modular forms we described above, which are not modular functions, i.e. they have nonzero weight! Why is this so? Let us recall that an elliptic curve over $\mathbb{C}$ corresponds to a lattice. Acting on a basis of this lattice by an element of $\text{SL}_{2}(\mathbb{Z})$ changes the basis, but preserves the lattice. This will be reflected as “admissible changes of coordinates” in the Weierstrass equations, and also changes these modular forms associated to the elliptic curves even though the elliptic curves are still isomorphic. But they change in a predictable way, according to the definition of modular forms.

A modular form $f(\tau)$ is also called a cusp form if the limit of $f(\tau)$ is zero as the imaginary part of $\tau$ approaches infinity. We denote the set of cusp forms of weight $k$ by $\mathcal{S}_{k}(\text{SL}_{2}(\mathbb{Z})$ . They are a vector subspace of $\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z})$ and the graded ring formed by their direct sum for all $k$ , denoted $\mathcal{S}_{k}(\text{SL}_{2}(\mathbb{Z})$ , is an ideal of the graded ring $\mathcal{M}(\text{SL}_{2}(\mathbb{Z})$ . Cusp forms form a very important part of modern research, but we will not discuss them much in this introductory post and leave them for the future.

Let us now discuss congruence subgroups of $\text{SL}_{2}(\mathbb{Z})$ (we have also discussed this briefly in Shimura Varieties), so that we can define more general modular forms with respect to such a congruence subgroup instead of just $\text{SL}_{2}(\mathbb{Z})$ . Given an integer $N$ , the principal congruence subgroup $\Gamma(N)$ of $\text{SL}_{2}(\mathbb{Z})$ is the subgroup consisting of the elements which reduce to the identity when we reduce the entries modulo $N$ . A congruence subgroup is any subgroup $\Gamma$ that contains the principal congruence subgroup $\Gamma(N)$ . We refer to $N$ as the level of the congruence subgroup.

There are two important kinds of congruence subgroups of $\text{SL}_{2}(\mathbb{Z})$ , denoted by $\Gamma_{0}(N)$ and $\Gamma_{1}(N)$ . The subgroup $\Gamma_{0}(N)$ consists of the elements that become upper triangular after reduction modulo $N$ , while the subgroup $\Gamma_{1}(N)$ consists of the elements that become upper triangular with ones on the diagonal after reduction modulo $N$ . As we discussed in Shimura Varieties, these are related to moduli spaces of “elliptic curves with level structure”.

Now we can define the modular forms of weight $k$ with respect to such a congruence subgroup $\Gamma$ . We shall once again require them to be holomorphic functions on the upper half-plane, and we require that for $\gamma\in \Gamma$ written as $\left(\begin{array}{cc}a&b\\ c&d\end{array}\right)$ we must have

$\displaystyle f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)$ .

However, the condition that the function be bounded as the imaginary part of $\tau$ goes to infinity must be modified. The reason is that the “point at infinity” is a cusp, a point of the modular curve that does not correspond to an elliptic curve over $\mathbb{C}$ but rather to a “degeneration” of it (this point is therefore not a part of the usual moduli space of elliptic curves – we can think of it as a “puncture” in this space).

We recall that the construction of the moduli space of elliptic curves over $\mathbb{C}$ starts with the upper half-plane, then we quotient out by the action of $\text{SL}_{2}(\mathbb{Z})$ . The cusps come from taking the union of the rational numbers with the upper half-plane, as well as the point at infinity. When we take the quotient by $\text{SL}_{2}(\mathbb{Z})$ this all gets sent to the same point, therefore the usual moduli space has only one cusp. But if we take the quotient by a congruence subgroup, we may have several cusps. Therefore, what we really require is for the modular form to be “holomorphic at the cusps“. We can still express this condition in familiar terms by requiring that not $f(\tau)$ , but rather $(c\tau+d)^{-k}f(\gamma(\tau))$ for $\gamma\in \text{SL}_{2}(\mathbb{Z})$ be bounded as the imaginary part of $\tau$ goes to infinity. We can then define cusp forms with respect to $\Gamma$ by requiring vanishing at the cusps instead. The set of modular forms (resp. cusp forms) of weight $k$ with respect to $\Gamma$ are denoted $\mathcal{M}_{k}(\Gamma)$ (resp. $\mathcal{S}_{k}(\Gamma)$ ), and they also have the same structures of being vector spaces and being graded pieces of graded rings as the ones for $\text{SL}_{2}(\mathbb{Z})$ .

Having only discussed the very basics of modular forms we end the post here, with the hope that in the near future we will be able to discuss things such as Hecke operators, modular curves and their Jacobians, and their associated Galois representations. We redirect the interested reader to the references for now.

References:

Modular Form on Wikipedia

Eisenstein Series in Wikipedia

j-invariant on Wikipedia

Modular Form on Wikipedia

Congruence Subgroups on Wikipedia

Image by User Fropuff of Wikipedia

A First Course in Modular Forms by Fred Diamond and Jerry Shurman

Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman