Arithmetic local constants for abelian varieties with extra endomorphisms

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.