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
are congruent to
the traces of elements ρ(Frobp), where Frobp are certain special
elements of GQ called Frobenius elements. See [RS01] and [DS05,
(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 ⊂
is an even unimodular lattice (the basis matrix
has determinant ±1 and λ · λ ∈ 2Z for all λ ∈ L), then the theta
is a modular form of weight n/2. The coeﬃcient of
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).
1.1 Suppose γ =
∈ 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.