Abstract

. This is a introduction to some aspects of the arithmetic of elliptic curves, intended for readers with little or no background in number theory and algebraic geometry. In keeping with the rest of this volume, the presentation has an algorithmic slant. We also touch lightly on curves of higher genus. Readers desiring a more systematic development should consult one of the references for further reading suggested at the end. 1. Plane curves Let k be a field. For instance, k could be the field Q of rational numbers, the field R of real numbers, the field C of complex numbers, the field Q p of p-adic numbers (see [Kob] for an introduction), or the finite field F q of q elements (see Chapter I of [Ser1]). Let k be an algebraic closure of k. A plane curve 1 X over k is defined by an equation f(x, y) = 0 where f(x, y) = # a ij x i y j # k[x, y] is irreducible over k. One defines the degree of X and of f by deg X = deg f = max{i + j : a ij #= 0}. A k-rational point (or simply k-point) on X is a point (a, b) with coordinates in k such that f(a, b) = 0. The set of all k-rational points on X is denoted X(k). Example: The equation x 2 y - 6y 2 - 11 = 0 defines a plane curve X over Q of degree 3, and (5, 1/2) # X(Q). Already at this point we can state an open problem, one which over the centuries has served as motivation for the development of a huge amount of mathematics. Question. Is there an algorithm, that given a plane curve X over Q, determines X(Q), or at least decides whether X(Q) is nonempty? Although X(Q) need not be finite, we will see later that it always admits a finite description, so this problem of determining X(Q) can be formulated precisely using the notion of Turing machine: see [HU] for a definition. For the relationship of this questi...

Citations