The average size of the 3-isogeny Selmer groups of elliptic curves $y^2 = x^3 + k$

TL;DR

This paper calculates the average size of 3-isogeny Selmer groups for a family of elliptic curves, revealing sensitivity to congruence conditions and implications for average rank and distribution of ranks.

Contribution

It provides the first precise computation of the average size of 3-isogeny Selmer groups for curves of the form y^2 = x^3 + k, highlighting the influence of congruence conditions and Tamagawa numbers.

Findings

Average 3-isogeny Selmer group size is computed.

Average rank of curves in the family is less than 1.21.

Significant proportions of curves have rank 0 and rank 3-Selmer rank 1.

Abstract

The elliptic curve $E_k \colon y^2 = x^3 + k$ admits a natural 3-isogeny $\phi_k \colon E_k \to E_{-27k}$. We compute the average size of the $\phi_k$-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 $E_{-27k}$. As consequences, we prove that the average rank of the curves $E_k$, $k\in\mathbb Z$, is less than 1.21 and over $23\%$ (resp. $41\%$) of the curves in this family have rank 0 (resp. 3-Selmer rank 1).