Multi-Page Printing

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,