# The Arithmetic Theory of Local Constants for Abelian Varieties

### Marco Adamo Seveso

Università degli Studi di Milano, Italy

## 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 $L$.

## Cite this article

Marco Adamo Seveso, The Arithmetic Theory of Local Constants for Abelian Varieties. Rend. Sem. Mat. Univ. Padova 127 (2012), pp. 17–39

DOI 10.4171/RSMUP/127-2