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 ${\mathrm{End}<!--FUNCTION\; APPLICATION-->}_{\overline{F}}A$, 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 result 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.