A Comparison of Direct and Indirect Methods for Computing Selmer Groups of an Elliptic Curve

In this paper we examine diierences between the two standard methods for computing the 2-Selmer group of an elliptic curve. In particular we focus on practical diierences in the timings of the two methods. In addition we discuss how to proceed if one fails to determine the rank of the curve from the 2-Selmer group. Finally we mention brieey ongoing research i n to generalizing such methods to the case of computing the 3-Selmer group. Computing the 2-Selmer group is a basic problem in the computational theory of elliptic curves over the rationals. It is, assuming the Tate-Shaferevich group, X, has no 2-primary part, the most eecient way known of computing the rank and generators of the Mordell-Weil group. That we do not have a n a l-gorithm to compute the Mordell-Weil group in general is one of the major open problems in the theory of elliptic curves. The computation of the Mordell-Weil group is basic to many Diophantine problems such as computing the set of integral points on a curve via elliptic logarithms, 11, 222, 21, or verifying the Birch-Swinnerton-Dyer conjecture, 2, 33. Throughout this paper, by an elliptic curve w e shall mean a curve of the form 1 where I ; J2Z Z. W e let = 4 I 3 , J 2 denote the discriminant of the curve. There are currently two methods used to compute the 2-Selmer group, S 2. The rst method, which is essentially part of the standard proof of the Mordell-Weil theorem, uses number eld arithmetic. This method works directly with the Selmer group and can therefore make explicit use of the underlying group structure of the elements. The second method, due to Birch and Swinnerton-Dyer, 22, computes S 2 in an indirect way b y computing a set of binary quartic forms which indirectly represent the elements of S 2 .