Computing the p-Selmer Group of an Elliptic Curve

In this paper we explain how to bound the p-Selmer group of an elliptic curve over K, a number eld. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number elds of degree at most p 1. Our method is practical for p = 3 but for larger values of p becomes impractical with current computing power. In the examples we have calculated, our method produces exactly the p-Selmer group of the curve, and so one can use the method to nd the Mordell-Weil rank of the curve when the usual method of 2-descent fails. Let E denote an elliptic curve, de ned over K, a number eld, which we assume is given by an equation of the form y = x + ax+ b: It is a fundamental question as to how to determine the Mordell-Weil rank of such curves. The standard method is to perform a descent type argument and to bound the size of E(K)=mE(K) for some integer m. This is usually done by determining the mth Selmer group, Sm(E=K) which contains E(K)=mE(K). In the literature this method is only completely explained in the case m = 2, see [3] and [4], for which very e cient programs exist to carry out the computation, for example [7]. For other prime values ofm the method, as it is described, requires one to extend the ground eld to the minimal algebraic number eld, L, which contains them-division points, [18, VIII Lemma 1.1.1]. The Selmer group, Sm(E=L), is then determined and not Sm(E=K). The structure of Sm(E=K), and hopefully generators of E(K), can then be obtained using a Galois descent argument. This is unsatisfactory for several reasons. If we rely on the use only of m = 2, then we lay ourselves open to the problem that we may not be able to determine the rank as the curve may have a non-trivial 2-part of the Tate-Shafarevich group, X(E=K). Further descents, [10], can then be carried out but again this is only completely explicit for the case m = 4. Alternatively one can make use of the Cassels-Tate pairing onX(E=K), [5]. Extending to a number eld L also gives rise to problems. For exampleX(E=L) could have a nontrivial m-part whenX(E=K) does not. So we could in fact make matters worse by extending the ground eld. It is also somewhat against the spirit of the exercise to compute the generators of E(K) given the generators of E(L). In this paper we hope to remedy the situation by giving an explicit method to compute an approximation to the m-Selmer group, Sm(E=K), of an elliptic curve over K, for values of m which are prime. For the rest of the paper we shall take m = p, a prime number. We shall not need to extend the eld of de nition of the 1991 Mathematics Subject Classi cation. Primary: 11G05, 11Y99, Secondary: 14H52, 14Q05.