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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