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
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