Selmer groups of abelian varieties with cyclotomic multiplication

Abstract: Let $A$ be an abelian variety over a number field $F$, with complex multiplication by the $n$-th cyclotomic field $\mathbb{Q}(\zeta)$.  If $n = 3^m$, we show that the average size of the $(1-\zeta)$-Selmer group of $A_d$, as $A_d$ varies through the twist family of $A$, is equal to 2.  As a corollary, the average $\mathbb{Z}[\zeta]$-rank of $A_d$ is at most 1/2, and at least 50% of $A_d$ have rank 0.    More generally, we prove average rank bounds for various twist families of abelian varieties with "cyclotomic" multiplication (not necessarily CM) over $\bar F$, such as sextic twist families of trigonal Jacobians over $\mathbb{Q}$. These results have application to questions of  "rank gain" for a fixed elliptic curve over a family of sextic fields, as well as the distribution of $\#C_d(F)$, as $C_d$ varies through twists of a fixed curve $C$ of genus $ g > 1$.  This is joint work with Ariel Weiss.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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