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This work generalizes the theory of arithmetic local constants, introduced byMazur and Rubin, to better address abelian varieties with a larger endomorphismring than $\mathbb{Z}$. We then study the growth of the $p^\infty$-Selmer rankof our abelian variety, and we address the problem of extending the results ofMazur and Rubin to dihedral towers $k\subset K\subset F$ in which $[F:K]$ isnot a $p$-power extension.

Arithmetic local constants for abelian varieties with extra endomorphisms