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• W3178219295 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." @default.
• W3178219295 created "2021-07-19" @default.
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• W3178219295 date "2021-07-14" @default.
• W3178219295 modified "2023-09-27" @default.
• W3178219295 title "Ranks of abelian varieties in cyclotomic twist families" @default.
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