Ranks of abelian varieties in cyclotomic twist families

Let A be an abelian variety over a number field F, and suppose that Z[ζₙ] embeds in EndF A, for some root of unity ζₙ of order n = 3ᵐ. Assuming that the Galois action on the finite group A[1-ζₙ] is sufficiently reducible, we bound the average rank of the Mordell--Weil groups Ad(F), as Ad varies through the family of μ₂ₙ-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 y³ = f(x²), as well as in twist families of theta divisors of cyclic trigonal curves y³ = f(x). Our main technical result is the determination of the average size of a $3$-isogeny Selmer group in a family of μ₂ₙ-twists.