Ranks of abelian varieties in cyclotomic twist families

Shnidman, Ari; Weiss, Ariel

doi:10.2140/ant.2025.19.39

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Ranks of abelian varieties in cyclotomic twist families

Ari Shnidman and Ariel Weiss

Vol. 19 (2025), No. 1, 39–75

DOI: 10.2140/ant.2025.19.39

Abstract

Let $A$ be an abelian variety over a number field $F$ , and suppose that $ℤ [ζ_{n}]$ embeds in ${End}_{\bar{F}} A$ , 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 $μ_{2 n}$ -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 $μ_{2 n}$ -twists.

Keywords

arithmetic statistics, twist families, rational points on curves, ranks of abelian varieties

Mathematical Subject Classification

Primary: 11G10

Secondary: 11E76, 11S25, 14G05

Milestones

Received: 27 May 2022

Revised: 12 December 2023

Accepted: 22 January 2024

Published: 4 December 2024

Authors

Ari Shnidman
Einstein Institute of Mathematics The Hebrew University of Jerusalem Jerusalem Israel

Ariel Weiss
Department of Mathematics The Ohio State University Columbus, OH United States	Department of Mathematics Ben-Gurion University of the Negev Be’er Sheva Israel

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Vol. 19, No. 1, 2025