Ranks of abelian varieties in cyclotomic twist families

Abstract

Let $A$ be an abelian variety over a number field $F$, and suppose that $\mathbb Z[\zeta_n]$ embeds in $\mathrm{End}_{\bar F} A$, for some root of unity $\zeta_n$ of order $n = 3^m$. Assuming that the Galois action on the finite group $A[1-\zeta_n]$ is sufficiently reducible, we bound the average rank of the Mordell--Weil groups $A_d(F)$, as $A_d$ varies through the family of $\mu_{2n}$-twists of $A$. Combining this with the recently proved uniform Mordell--Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves $y^3 = f(x^2)$, as well as in twist families of theta divisors of cyclic trigonal curves $y^3 = f(x)$. Our main technical result is the determination of the average size of a $3$-isogeny Selmer group in a family of $\mu_{2n}$-twists.