Rational Points on Curves and their Jacobians

We will survey methods for computing the Mordell-Weil group E(K) of an elliptic curve defined over a number field K, concentrating on the case K=Q. The theoretical result that E(K) is finitely-generated is proved in two stages, first showing that E(K)/mE(K) is finite for some m>1 and then deducing that E(K) is finitely-generated using the theory of heights. Similarly, algorithms for computing E(K) first find E(K)/mE(K), using a so-called m-descent. This determines the m-Selmer group of E, whose size gives an upper bound for the rank of E(K). The simplest case to make explicit and to implement is when m=2, and we will describe the method of 2-descent in some detail. The second stage of the algorithm, which consists of determining generators for E(K) from generators of E(K)/mE(K), involves a sieving procedure together with estimates from the theory of heights. [All the algorithms we describe have been, or are in the process of being, implemented in Magma.]