Arithmetic local constants for abelian varieties with extra endomorphisms

TL;DR

This paper extends the theory of arithmetic local constants to abelian varieties with larger endomorphism rings and investigates the growth of their Selmer ranks in complex field extensions.

Contribution

It generalizes the concept of local constants to a broader class of abelian varieties and studies Selmer rank growth in non-p-power dihedral towers.

Findings

Generalized local constants for abelian varieties with extra endomorphisms.

Analyzed Selmer rank growth in complex field extensions.

Extended Mazur-Rubin results to non-p-power dihedral towers.

Abstract

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than $\mathbb{Z}$. We then study the growth of the $p^\infty$-Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers $k\subset K\subset F$ in which $[F:K]$ is not a $p$-power extension.