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
n1
(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
/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.