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: Description: Research/departmental seminar

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