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