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

Abstract

The elliptic curve $E_k:y^2=x^3+k$ admits a natural 3-isogeny ${\varphi }_k:E_k\to E_{-27k}$. We compute the average size of the ${\varphi }_k$-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).

25 pages

References in corpus (4)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
• Many cubic surfaces contain rational points
• The average number of integral points on elliptic curves is bounded

Cited by in corpus (9)

• Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families
• Many cubic surfaces contain rational points
• Rank growth of elliptic curves over $N$-th root extensions
• Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$
• Ranks of abelian varieties in cyclotomic twist families
• Vanishing criteria for Ceresa cycles
• The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$
• Remarks on the Birch-Swinnerton-Dyer conjecture
• On certain root number $1$ cases of the cube sum problem