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, with emphasis to abelian varieties with real multiplication. Let $l$ be an odd rational prime and let $L/K$ be an abelian $l$-power extension. Assume that we are given a quadratic extension $K/k$ such that $L/k$ is a dihedral extension and the abelian variety $A/k$ is defined over $k$ and polarizable. This theory can be used to relate the rank of the $l$-Selmer group of $A$ over $K$ to the rank of the $l$-Selmer group of $A$ over $L$.