How to do a 𝑝-descent on an elliptic curve

In this paper, we describe an algorithm that reduces the computation of the (full) p p -Selmer group of an elliptic curve E E over a number field to standard number field computations such as determining the ( p p -torsion of) the S S -class group and a basis of the S S -units modulo p p th powers for a suitable set S S of primes. In particular, we give a result reducing this set S S of ‘bad primes’ to a very small set, which in many cases only contains the primes above p p . As of today, this provides a feasible algorithm for performing a full 3 3 -descent on an elliptic curve over Q Q , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of E [ p ] E[p] is favorable, simplifications are possible and p p -descents for larger p p are accessible even today. To demonstrate how the method works, several worked examples are included.