On the Universal Norms of Some Abelian Varieties Over Local Fields

Abstract

Let F be a finite extension field of Q_p , A an abelian variety defined over F with ordinary good reduction and with sufficiently many endomorphisms (see the theorem below for a precise statement). In this paper we prove that there exists unique Galois extension M of F such that for a Galois extension K of F, the group N_K ⁄F (A ) of universal norms is finite if and only if K contains M. Our result generalizes that of J. Coates and R. Greenberg [1] which concerns the case of elliptic curves.