Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field

TL;DR

This paper proves that the average rank of quadratic twists of certain abelian varieties over number fields is bounded, advancing understanding of ranks in twist families and providing new evidence related to Goldfeld's conjecture.

Contribution

It establishes the boundedness of average ranks for quadratic twists of abelian varieties with specific isogeny factorizations, including new results for higher dimensions and over arbitrary number fields.

Findings

Average rank of quadratic twists is bounded under certain isogeny conditions.

Positive proportions of twists have rank 0 and 3-Selmer rank 1 over totally real fields.

Progress towards Goldfeld's conjecture over general number fields.

Abstract

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension one, we deduce that if $E/F$ is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If $F$ is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have $3$-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld's conjecture -- which states that the average rank in the quadratic twist family of an elliptic curve over $\mathbb{Q}$ should be $1/2$ -- and the first progress towards the analogous conjecture over number fields other than $\mathbb{Q}$. Our results follow from a computation of the average size of the $\phi$-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny $\phi$.