Introduction to Computing With Modular Forms

William A. Stein was@math.harvard.edu http://modular.fas.harvard.edu/∼was

8 March 2001 at Arizona State University

1 Some open questions

Here are two central questions in number theory that are, for the most part, completely open today:

1. The Birch and Swinnerton-Dyer conjecture: a simple conjectural criterion that decides whether or not a cubic equation y^{2} = x^{3} + ax + b has infinitely many solutions.

2. Serre’s conjecture: Classify the odd 2-dimensional mod-ℓ representations of Gal(\overline{Q}/Q).

We now have partial solutions to each of these problems, and each involves modular forms.

2 What is a modular form?

Let N and k be positive integers.

Definition 2.1 (Modular form). A modular form of level N and weight k is a holomorphic function on the (extended) upper half plane such that for every determinant-1 matrix ( \begin{matrix}a & b \\ c & d \end{matrix}) with N | c we have

f (\frac{az + b}{cz + d}) = (cz + d)^{k}f(z).

We will denote the complex vector space of these modular forms by M_{k}(N) = M_{k}(Γ_{0}(N)).

Example 2.2. The Eisenstein series

E_{k} = ∑ \frac{1}{(nz + m)^{k}} ,

for k ≥ 4 an even integer, is a modular form of level 1 and weight k. (Here we sum over all pairs (n, m) ∈ Z × Z with not both n and m equal to zero.) Notice that

E_{k} (\frac{az + b}{cz + d}) = ∑ \frac{1}{(n\frac{az+b}{cz+d} + m)^{k}}

= ∑ \frac{(cz + d)^{k}}{(n(az + b) + m(cz + d))^{k}}

= (cz + d)^{k} ∑ \frac{1}{((na + mc)z + (nb + md))^{k}} = (cz + d)^{k}E_{k}(z)

(I won’t check holomorphicity.)

3 Computing with modular forms

3.1 Representation of modular forms

Because ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) has lower left entry divisible by N, every modular form f has a Fourier expansion

f(z) = ∑_{n≥0} a_{n}q^{n},

where q(z) = e^{2πiz}. In cases of interest, the a_{n} lie in a fixed finite extension of Q. We usually give a modular forms by giving its weight, level, and the first few terms of its Fourier expansion. E.g.,

E_{4} = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n} and E_{6} = 1 − 504 ∑_{n=1}^{∞} σ_{5}(N)q^{n}.

3.2 Generating the space of modular forms

Modular forms of level 1 are relatively simple to compute using any basic computer algebra system.

Theorem 3.1. A basis for M_{k}(1) is

{E_{4}^{i}E_{6}^{j} : 4i + 6j = k, and i, j ≥ 0}.

However, this method fails miserably when N > 1 because there aren’t enough Eisenstein series. For example, M_{2}(11) has dimension two but there is only one Eisenstein series in M_{2}(11). However, one can compute the forms of low weight and then generate the forms of higher weight by multiplying the low weight forms together.

3.3 The Hecke algebra

There is a commutative ring

T = Z[… T_{n} …]

of endomorphisms of M_{k}(N). It has the property that T_{n}(f) = a_{n}(f)=nth coefficient of f; moreover, M_{k}(N) = Hom(T, C). Thus to compute M_{k}(N) it suffices to compute T. Example 3.2. There is a basis for M_{2}(11) such that

T_{1} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , T_{2} = (\begin{matrix}−2 & 0 \\ 1 & 3 \end{matrix}) , T_{3} = (\begin{matrix}−1 & 0 \\ 1 & 4 \end{matrix}) , T_{4} = (\begin{matrix}2 & 0 \\ 1 & 7 \end{matrix}) , T_{5} = (\begin{matrix}1 & 0 \\ 1 & 6 \end{matrix}) .

From this we deduce that the following two modular forms span M_{2}(11):

f_{1} = q − 2q^{2} − q^{3} + 2q^{4} + q^{5} + ⋯

and

f_{2} = q + 3q^{2} + 4q^{3} + 7q^{4} + 6q^{5} + · · · .

By counting points on the elliptic curve E given by y^{2}+y = x^{3}−x^{2}−10x−20 of conductor 11 one finds the following modular form:

f_{E} = q − 2q^{2} − q^{3} + 2 ∗ q^{4} + q^{5} + · · · .

Observe that f_{E} = f_{1}.

The Hecke algebra T acts on many spaces in addition to the space of modular forms. For example, T acts on the space of modular symbols, which is a very explicitly presented complex vector space. By reading off the T action on modular symbols, one recovers the q-expansions of modular forms.

3.4 Modular symbols

When k = 2 and N = p is prime, the space of modular symbols is presented as follows. It is the quotient of the vector space generated by symbols [0], [1], ..., [p − 1], [∞] by the relations

x + xσ = 0 and x + xτ + xτ^{2} = 0,

where σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and τ = ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ). For more details, see my paper An introduction to computing modular forms using modular symbols.

3.5 Quadratic forms

Using the ideal theory of queternion algebras, it is possible to list a collection of θ series ∑(num ways to represent n)q^{n} that generate certain spaces of modular forms. In the case of weight two, David Kohel implemented this algorithm in Magma.

3.6 Hecke

I wrote a package called “HECKE” that comes with Magma, which uses modular symbols for most internal computations. It includes functions that, e.g., compute a basis of q-expansions for the subspace of modular forms that are “0 at the cusps”.

[[If the audience is small, show them!]]

4 Two applications

4.1 The Birch and Swinnerton-Dyer conjecture

Birch and Swinnerton-Dyer made a conjecture about the behaviour at s = 1 of a the L-function attached to an abelian variety. It was recently proved by Wiles and others that every cubic equation y^{2} = x^{3} + ax + b is associated to a modular form. For cubic curves associated to modular forms, quite a lot of the BSD conjecture is known.

4.2 Serre’s Conjecture

Serre conjected that odd irreducible two-dimensional modular representations of G = Gal(\overline{Q}/Q) are “attached to modular forms” (NOT proved!) Moreover, he gave a very precise recipe for the space of modular forms that should give rise to the representation (mostly proved!). Probably about half of the explicit modular forms computations I’ve encountered have involved this amazingly useful conjecture. Also, the “recipe” part of Serre’s conjecture was used in the proof of FLT.