Ranks of abelian varieties in cyclotomic twist families

Let A be an abelian variety over a number field F, and suppose that [ n ] embeds in End F A, for some root of unity n of order n = 3 m .Assuming that the Galois action on the finite group A[1 - 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 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 2n -twists.X avg d X rk A d (F).