1.6. Exercises 11

(8) Serre’s Conjecture on modularity of Galois representation:

Let GQ = Gal(Q/Q) be the Galois group of an algebraic closure

of Q. Serre conjectured and many people have (nearly!) proved

that every continuous homomorphism ρ : GQ → GL2(Fq), where

Fq is a finite field and det(ρ(complex conjugation)) = −1, “arises”

from a modular form. More precisely, for almost all primes p the

coeﬃcients ap of a modular (eigen-)form

∑

anqn

are congruent to

the traces of elements ρ(Frobp), where Frobp are certain special

elements of GQ called Frobenius elements. See [RS01] and [DS05,

Ch. 9].

(9) Generating functions for partitions: The generating functions

for various kinds of partitions of an integer can often be related to

modular forms. Deep theorems about modular forms then translate

into results about partitions. See work of Ramanujan, Gordon,

Andres, and Ahlgren and Ono (e.g., [AO01]).

(10) Lattices: If L ⊂

Rn

is an even unimodular lattice (the basis matrix

has determinant ±1 and λ · λ ∈ 2Z for all λ ∈ L), then the theta

series

θL(q) =

λ∈L

qλ·λ

is a modular form of weight n/2. The coeﬃcient of

qm

is the num-

ber of lattice vectors with squared length m. Theorems and com-

putational methods for modular forms translate into theorems and

computational methods for lattices. For example, the 290 theorem

of M. Bharghava and J. Hanke is a theorem about lattices, which

asserts that an integer-valued quadratic form represents all posi-

tive integers if and only if it represents the integers up to 290; it

is proved by doing many calculations with modular forms (both

theoretical and with a computer).

1.6. Exercises

1.1 Suppose γ =

(

a b

c d

)

∈ GL2(R) has positive determinant. Prove that

if z ∈ C is a complex number with positive imaginary part, then

the imaginary part of γ(z) = (az + b)/(cz + d) is also positive.

1.2 Prove that every rational function (quotient of two polynomials) is

a meromorphic function on C.

1.3 Suppose f and g are weakly modular functions for a congruence

subgroup Γ with f = 0.

(a) Prove that the product fg is a weakly modular function for Γ.

(b) Prove that 1/f is a weakly modular function for Γ.