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 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