The Arithmetic Theory of Local Constants for Abelian Varieties

Abstract

We present a generalization of the theory of local constants 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.

Arithmetic local constants for abelian varieties with extra endomorphisms

“…We consider the ring of integers O of a number field K, and assume O ⊂ End K (X) is contained in the ring of endomorphisms of X defined over K. The case O = Z and K = Q is that of Mazur and Rubin in [9]. Recent work of Seveso [15] addresses similar questions for abelian varieties with real multiplication.…”

Arithmetic local constants for abelian varieties with extra endomorphisms

“…We consider the ring of integers O of a number eld K, and assume O ⊂ End K (X) is contained in the ring of endomorphisms of X dened over K. The case O = Z and K = Q is that of Mazur and Rubin in [9]. Recent work of Seveso [15] addresses similar questions for abelian varieties with real multiplication.…”