Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

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Classical Modular Forms

Introduction

William Stein

Department of Mathematics University of Washington

August 31, 2006 / MSRI Workshop

Outline

1 Definition of Modular Forms (of Level 1)

2 Applications of Modular Forms

3 Workshop Goals

Definition of Modular Forms (of Level 1)

Applications of Modular Forms

Workshop Goals

Some Objects

Definition (Upper Half Plane) The complex upper half plane is 𝔥 = {z ∈ ℂ : Im(z) > 0}.

Definition (Modular Group) The modular group

SL_{2}(ℝ) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) : ad − bc = 1, and a, b, c, d ∈ ℝ\}

acts on 𝔥 via linear fractional transformations.

Definition (Congruence Subgroup) A congruence subgroup Γ of SL_{2}(ℤ) is one that contains Γ(N) = ker(SL_{2}(ℤ) → SL_{2}(ℤ/Nℤ)) for some N.

William Stein  Classical Modular Forms

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Weakly modular functions

Definition (Weakly Modular Function for Γ) A weakly modular function of weight k ∈ ℤ for Γ is a meromorphic function f on 𝔥 such that for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ and all z ∈ 𝔥 we have

f (z) = (cz + d)^{−k}f (γ(z)).  (1.1)

When k is even:

f (γ(z))(d(γ(z)))^{k/2} = f (z)(dz)^{k/2}.

In this case (1.1) says that the weight k “differential form” f (z)(dz)^{k/2} is fixed under the action of every element of Γ.

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Fourier Expansions

Suppose f is a weakly modular function of weight k for Γ. Then f (z) = f (z + h) for some integer h > 0.

Definition (Fourier Expansion) A Fourier expansion of f , if it exists, is a representation of f as f (z) = ∑_{n=m}^{∞} a_{n}e^{2πin/hz}, for all z ∈ 𝔥.

The Fourier expansion of f at a cusp α ∈ Γ\ℙ^{1}(ℚ) is the Fourier expansion of f^{[γ]_{k}} at ∞, for some γ ∈ SL_{2}(ℤ) such that γ(∞) = α. One of the problems for this workshop is to investigate efficient computation of these expansions at other cusps. We very often compute with Fourier expansions to some precision.

William Stein  Classical Modular Forms

Definition of Modular Forms (of Level 1)

Applications of Modular Forms

Workshop Goals

Modular Forms Are...

Definition (Modular Function) A modular function of weight k for Γ is a weakly modular function of weight k that is meromorphic at all cusps.

Definition (Modular Form) A modular form of weight k for Γ is a modular function of weight k for Γ that is holomorphic on 𝔥 and at all cusps.

Definition (Cusp Form) A cusp form of weight k for Γ is a modular form of weight k such that f (α) = 0 for all cusps α.

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Are Modular Forms Interesting?

The definition of modular forms might leave the impression that modular forms occupy an obscure corner of complex analysis. This is not the case! Modular forms are highly geometric, arithmetic, and topological objects that are of extreme interest all over mathematics, as the following examples illustrate.

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Fermat’s last theorem

Wiles’s proof of Fermat’s last theorem uses modular forms extensively. The work of Wiles, Taylor, etc., on modularity of elliptic curves also massively extends computational methods for elliptic curves over ℚ, because many elliptic curve algorithms, e.g., for computing L-functions, modular degrees, Work of Kolyvagin, Kato, etc., on the Birch and Swinnerton-Dyer conjecture all uses modularity extensively. E.g., Construction of Heegner points require that the elliptic curve is modular.

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Diophantine equations

Andre Wiles’s proof of Fermat’s last theorem has made available a wide array of new techniques for solving certain diophantine equations. Such work relies crucially on having access to tables or software for computing modular forms. Wiles did not need a computer, because the relevant spaces of modular forms that arise in his proof have dimension 0! Also, according to Samir Siksek (personal communication) some of his papers about diophantine equations would “have been entirely impossible to write without the algorithms you’ll talk about.”

(I’ll talk about modular symbols algorithms for computing modular forms in my lecture 2.)

William Stein  Classical Modular Forms

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Congruent number problem

Give an algorithm to decide whether an integers is the area of a right triangle with rational sides lengths. This thousand year old open problem is to determine which integers are the area of a right triangle with rational side lengths. There is a “potential” solution that uses modular forms (of weight 3/2) extensively (the solution is conditional on truth of the Birch and Swinnerton-Dyer conjecture, which is not yet known).

Topology

Topological modular forms are a major area of research (see work of Mike Hopkins et al.)...

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Construction of Ramanujan graphs

Modular forms can be used to construct almost optimal expander graphs, which play a role in communications network theory. Lassina Dembele and David Kohel’s lectures will likely be very relevant to this problem.

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Cryptography and Coding Theory

Point counting on elliptic curves over finite fields is crucial to the construction of elliptic curve cryptosystems. Modular forms are very relevant to such algorithms. Algebraic curves that are associated to modular forms are useful in constructing error-correcting codes.

Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

The Birch and Swinnerton-Dyer conjecture

This central open problem in arithmetic geometry relates arithmetic properties of elliptic curves (and abelian varieties) to special values of L-functions. Most deep results toward this conjecture use modular forms extensively (e.g., work of Kolyvagin, Gross-Zagier, Kato, Skinner-Urban, etc.). Also, modular forms are used to compute and prove results about special values of these L-functions.

(I’ll talk about computational verification of the Birch and Swinnerton-Dyer conjecture in my lecture 4.)