Ranks of abelian varieties in cyclotomic twist families

Let A be an abelian variety over a number field F, and suppose that Z[ζ_n] embeds in (Formula Presented), for some root of unity ζ_n of order n = 3^m. Assuming that the Galois action on the finite group A[1 − ζ_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 µ_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 y³ = f (x²), as well as in twist families of theta divisors of cyclic trigonal curves y³ = f (x). Our main tech-nical result is the determination of the average size of a 3-isogeny Selmer group in a family of µ_2n-twists.