0Introduction

III Modular Forms and L-functions

0 Introduction

One of the big problems in number theory is the so-called Langlands programme,

which is relates “arithmetic objects” such as representations of the Galois group

and elliptic curves over

Q

, with “analytic objects” such as modular forms and

more generally automorphic forms and representations.

Example. y

2

+

y

=

x

3

− x

is an elliptic curve, and we can associate to it the

function

f(z) = q

Y

n≥1

(1 − q

n

)

2

(1 − q

11n

)

2

=

∞

X

n=1

a

n

q

n

, q = e

2πiz

,

where we assume

Im z >

0, so that

|q| <

1. The relation between these two

objects is that the number of points of

E

over

F

p

is equal to 1 +

p − a

p

, for

p 6

= 11. This strange function

f

is a modular form, and is actually cooked up

from the slightly easier function

η(z) = q

1/24

∞

Y

n=1

(1 − q

n

)

by

f(z) = (η(z)η(11z))

2

.

This function

η

is called the Dedekind eta function, and is one of the simplest

examples of a modular forms (in the sense that we can write it down easily).

This satisfies the following two identities:

η(z + 1) = e

iπ/12

η(z), η

−1

z

=

r

z

i

η(z).

The first is clear, and the second takes some work to show. These transformation

laws are exactly what makes this thing a modular form.

Another way to link E and f is via the L-series

L(E, s) =

∞

X

n=1

a

n

n

s

,

which is a generalization of the Riemann ζ-function

ζ(s) =

∞

X

n=1

1

n

s

.

We are in fact not going to study elliptic curves, as there is another course

on that, but we are going to study the modular forms and these

L

-series. We are

going to do this in a fairly classical way, without using algebraic number theory.