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

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 $\varphi$-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny $\varphi$.

References in corpus (2)

• Arithmetic aspects of the Burkhardt quartic threefold
• Quadratic twists of abelian varieties and disparity in Selmer ranks

Cited by in corpus (7)

• The average size of the 3-isogeny Selmer groups of elliptic curves $y^2=x^3+k$
• On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes
• Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families
• Rank growth of elliptic curves over $N$-th root extensions
• Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$
• Genus two curves with full $\sqrt{3}$-level structure and Tate-Shafarevich groups
• Ranks of abelian varieties in cyclotomic twist families