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,
