  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.
