2.1. Why elliptic curves? 19

Finding integral and rational points on a conic is a classical

problem. Legendre’s criterion determines whether there are

rational solutions: a conic C has rational solutions if and

only if C has points over R and over Qp, the p-adics, for all

primes p ≥ 2 (see Appendix D for a brief introduction to

the p-adics). Essentially, Legendre’s criterion says that the

conic has rational solutions if and only if there are solutions

modulo

pn

for all primes p and all n ≥ 1 but, in practice,

one only needs to check this for a finite number of primes

that depends on the coeﬃcients of the conic.

If C has rational points, and we have found at least

one point, then we can find all the rational solutions using

a stereographic projection (see Exercise 2.12.2). The inte-

gral points on C, however, are much more diﬃcult to find.

The problem is equivalent to finding integral solutions to

Pell’s equation x2 − Dy2 = 1. There are several methods to

solve Pell’s equation. For example, one can use continued

fractions (certain convergents

x

y

of the continued fraction

for

√

D are integral solutions (x, y) of Pell’s equation; see

Exercise 2.12.2).

• Cubic equations:

aX3

+

bX2Y

+ cXY

2

+ dY

3

+

eX2

+ fXY + gY

2

+ hX + jY + k = 0.

A cubic equation in two variables may have no rational solu-

tions, only 1 rational solution, a finite number of solutions,

or infinitely many solutions. Unfortunately, we do not know

any algorithm that yields all rational solutions of a cubic

equation, although there are conjectural algorithms. In this

chapter we will concentrate on this type of equation: a non-

singular cubic, i.e., no self-intersections or pinches, with at

least one rational point (which will be our definition of an

elliptic curve).

• Higher degree. Typically, curves defined by an equation of

degree ≥ 4 have a genus ≥ 2 (but some equations of degree

4 have genus 1; see Example 2.2.5 and Exercise 2.12.4). The

genus is an invariant that classifies curves according to their

topology. Briefly, if we consider a curve as defined over C,