The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

Abstract

The elliptic curve E k : y 2 = x 3 + k admits a natural 3-isogeny ϕ k : E k → E − 27 k . We compute the average size of the ϕ k -Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k; this sensitivity can be precisely controlled by the Tamagawa numbers of E k and E − 27 k . As a consequence, we prove that the average rank of the curves E k , k ∈ Z , is less than 1.21 and over 23 % (respectively, 41 % ) of the curves in this family have rank 0 (respectively, 3-Selmer rank 1).