A First Course in Modular Forms

This chapter introduces three central objects of the book.

For any congruence subgroup

Γ

of SL

2

(

Z

) the corresponding modular curve has been defined as the quotient space

$$ \Gamma \backslash \mathcal{H} $$

, the set of orbits

$$ Y{\text{(}}\Gamma {\text{) = }}\left\{ {\Gamma \tau :\tau \varepsilon \mathcal{H}} \right\}. $$

This chapter shows that

Y

(

Γ

) can be made into a Riemann surface that can be compactified. The resulting compact Riemann surface is denoted

X

(

Γ

).

For any congruence subgroup

Γ

of SL

2

(

Z

), the compactified modular curve

X

(

Γ

) is now a Riemann surface. The genus of

X

(

Γ

), its number of elliptic points, its number of its cusps, and the meromorphic functions and meromorphic differentials on

X

(

Γ

) are all used by Riemann surface theory to determine dimension formulas for the vector spaces

$$ \mathcal{M}_k (\Gamma ) $$

and

S

k

(

Γ

).

For any congruence subgroup

Γ

of SL

2

(

Z

), the space

$$ \mathcal{M}_k (\Gamma ) $$

of modular forms naturally decomposes into its subspace of cusp forms

S

k

(

Γ

) and the corresponding quotient space

$$ \mathcal{M}_k (\Gamma )/S_k (\Gamma ) $$

, the

Eisenstein space ε

k

(

Γ

). This chapter gives bases of

ε

k

(

Γ

(

N

)),

ε

k

(

Γ

1

(

N

)), and subspaces of

ε

k

(

Γ

1

(

N

)) called

eigenspaces

, including

ε

k

(

Γ

)

0

(

N

)). The basis elements are variants of the Eisenstein series from Chapter 1. For

k

≥ 3 they are straightforward to write down, but for

k

= 2 and

k

= 1 the process is different.

This chapter addresses the question of finding a canonical basis for the space of cusp forms

S

k

(

Γ

1

(

N

)). Since cusp forms are not easy to write explicitly like Eisenstein series, specifying a basis requires more sophisticated methods than the direct calculations of the preceding chapter.

Let

X

be a compact Riemann surface of genus

g

≥ 1 and fix a point

x

0

in

X

. Letting

x

vary over points of

X

and viewing path integration as a function of holomorphic differentials

ω

on

X

, the map

$$ x \mapsto \left( {\omega \mapsto \int_{x_0 }^x \omega } \right) $$

is an injection

$$ X \to \left\{ {\begin{array}{*{20}c} {{\text{linear functions of holomorphic differentials on }}X} \\ {{\text{modulo integration over loops in }}X} \\ \end{array} } \right\}. $$

When

g

= 1 this is an isomorphism of Abelian groups. When

g

< 1 the domain

X

is no longer a group, but the codomain still is. The codomain is the

Jacobian

of

X

, complex analytically a

g

-dimensional torus C

g

/

Λ

g

where

Λ

g

≅ Z

2

g

. This chapter presents the Jacobian and states a version of the Modularity Theorem mapping the Jacobian of a modular curve holomorphically to a given elliptic curve. The map is also a homomorphism, incorporating group structure into the Modularity Theorem whereas the first version, back in Chapter 2, was solely complex analytic. This chapter then uses the Jacobian to prove number-theoretic results about weight 2 eigenforms of the Hecke operators. It ends with another version of Modularity replacing the Jacobian with an

Abelian variety

, a quotient of the Jacobian. The Abelian variety comes from a weight 2 eigenform, so this version of the Modularity Theorem associates an eigenform to an elliptic curve.

Recall that every complex elliptic curve is described as an algebraic curve, the solution set of a polynomial equation in two variables, via the Weierstrass ℘-function,

$$ (\wp ,\wp '):C/\Lambda \to \{ (x,y):y^2 = 4x^3 - g_2 (\Lambda )x - g_3 \} \cup \{ \infty \} . $$

Let

N

be a positive integer. The modular curves

$$ X_0 (N) = \Gamma _0 (N)\backslash \mathcal{H}^* ,X_1 (N) = \Gamma _1 (N)\backslash \mathcal{H}^* ,X(N) = \Gamma (N)\backslash \mathcal{H}^* $$

can also be described as algebraic curves, solution sets of systems of polynomial equations in many variables. Since modular curves are compact Riemann surfaces, such polynomials with complex coefficients exist by a general theorem of Riemann surface theory, but

X

0

(

N

) and

X

1

(

N

) are in fact curves over the rational numbers, meaning the polynomials can be taken to have rational coefficients. The Modularity Theorem in its various guises rephrases with the relevant complex analytic objects replaced by their algebraic counterparts and with the relevant complex analytic mappings replaced by rational maps, i.e., maps defined by polynomials with rational coefficients. The methods of this chapter also show that

X

(

N

) is defined by polynomials with coefficients in the field

Q

(

μ

N

) obtained by adjoining the complex

N

th roots of unity to the rational numbers.

This chapter arrives at the first version of the Modularity Theorem stated in the preface to the book: For any elliptic curve

E

over

Q

there exists a newform

f

such that the Fourier coefficients

a

p

(

f

) are equal to the solution-counts

a

p

(

E

) of a Weierstrass equation for

E

modulo

p

. Gathering the

a

p

(

f

) and

a

p

(

E

) into

L

-functions, this rephrases as

$$ L(s,f) = L(s,E). $$

The techniques that relate this to other versions of Modularity involve working modulo

p

and expressing both

a

p

(

E

) and

a

p

(

f

) in terms of the

Frobenius map

,

$$ \sigma _p :x \mapsto x^p . $$

The key is the Eichler-Shimura relation, expressing the Hecke operator

T

p

in characteristic

p

in terms of

σ

p

. Since reducing algebraic curves and maps from characteristic 0 to characteristic

p

is technical, the chapter necessarily quotes many results in quickly sketching the relevant background. The focus here is on the Eichler-Shimura relation itself in Section 8.7 and on its connection to Modularity in Section 8.8.

This book has explained the idea that all elliptic curves over

Q

arise from modular forms. Chapters 1 and 2 introduced elliptic curves and modular curves as Riemann surfaces, and Chapter 1 described elliptic curves as algebraic curves over

C

. As a general principle, information about mathematical objects can be obtained from related algebraic structures. Elliptic curves already form Abelian groups. Modular curves do not, but Chapter 3 showed that the complex vector space of weight 2 cusp forms associated to a modular curve has dimension equal to the genus of the curve, Chapter 5 defined the Hecke operators, linear operators that act on the vector space, and Chapter 6 showed that integral homology is a lattice in the dual space and is stable under the Hecke action.