Computing the $p$-Selmer group of an elliptic curve

Abstract

In this paper we explain how to bound the $p$-Selmer group of an elliptic curve over a number field $K$. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number fields of degree at most $p^2-1$. Our method is practical for $p=3$, but for larger values of $p$ it 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 find the Mordell-Weil rank of the curve when the usual method of $2$-descent fails.