8 1. Introduction
f : H C that has several relations among its values (which we will
specify in Definitions 4.1.3 and 4.2.1). In particular, the values of the
function f satisfy several types of periodicity relations. For example,
the modular forms for SL(2, Z) satisfy, among other properties, the
following:
f(z) = f(z + 1) for all z H, and
f
(
−1
z
)
= zkf(z) for all z H. The number k is an integer
called the weight of the modular form.
We will describe modular forms in detail in Chapter 4. Let us see
some examples that motivate our interest in these functions.
Example 1.2.1 (Representations of integers as sums of squares). Is
the number n 0 a sum of two (integer) squares? In other words,
are there a, b Z such that n =
a2
+
b2?
And if so, in how many
different ways can you represent n as a sum of two squares?
For instance, the number n = 3 cannot be represented as a sum
of two squares but the number n = 5 has 8 distinct representations:
5 =
(±1)2
+
(±2)2
=
(±2)2
+
(±1)2.
Notice that here we consider
(−1)2 +22, 12 +22
and
22
+1 as distinct
representations of 5. A general formula for the number of represen-
tations of an integer n as a sum of 2 squares, due to Lagrange, Gauss
and Jacobi, is given by
S2(n) = 2 1 +
−1
n
d|n
−1
d
, (1.2)
where
(
m
n
)
is the Jacobi symbol and

d|n
is a sum over all positive
divisors of n (including 1 and n). Here we just need the easiest values
(
−1
n
) = (−1)(n−1)/2 of the Jacobi symbol. Let us see that the formula
works:
S2(3) = 2 1 +
−1
3
d|3
−1
d
= 2(1 + (−1))(1 + (−1)) = 0,
S2(5) = 2 1 +
−1
5
d|5
−1
d
= 2(1 + 1)(1 + 1) = 8,
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