The Arithmetic Theory of Local Constants for Abelian Varieties

Abstract

We present a generalization of the theory of local constant developed by B.Mazur and K. Rubin in order to cover the case of abelian varieties, withemphasis to abelian varieties with real multiplication. Let $l$ be an oddrational prime and let $L/K$ be an abelian $l$-power extension. Assume that weare given a quadratic extension $K/k$ such that $L/k$ is a dihedral extensionand the abelian variety $A/k$ is defined over $k$ and polarizable. This theorycan be used to relate the rank of the $l$-Selmer group of $A$ over $K$ to the rankof the $l$-Selmer group of $A$ over $L$.