Chapter 1 Introduction Notation: N = {1, 2,3,...} is the set of natural numbers. Z = {..., −3, −2, −1, 0,1,2,3,...} is the ring of integers. Q = { m n : m, n ∈ Z, n = 0} is the field of rational numbers. R is the field of real numbers. C = {a + bi : a, b ∈ R, i2 = −1} is the field of complex numbers. In this chapter, we introduce elliptic curves, modular forms and L- functions through examples that motivate the definitions. 1.1. Elliptic curves For the time being, we define an elliptic curve to be any equation of the form y2 = x3 + ax2 + bx + c with a, b, c ∈ Z and such that the polynomial x3 + ax2 + bx + c does not have repeated roots. See Section 2.2 for a precise definition. Example 1.1.1. Are there three consecutive integers whose product is a perfect square? There are some trivial examples that involve the number zero, for example, 0,1 and 2, whose product equals 0 · 1 · 2 = 0 = 02, a square. 1 http://dx.doi.org/10.1090/stml/058/01

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.