Selmer groups in twist families of elliptic curves

Abstract

The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute L-E(1). Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type alpha(log log(X))(1+epsilon)/log(X) with epsilon small. We discuss how the "best choice" of alpha is depending on the conductor of the chosen elliptic curves and the congruence classes of twist factors.