Galois actions on Tate modules of Abelian varieties with semi-stable reduction

Abstract

Let p be a rational prime number, let K denote a finite extension of $Q_p$, $\bar{K}$ some fixed algebraic closure of K. Let $G_K$ be the absolute Galois group of K and let $I_K\subset G_K$ be its inertial subgroup. Let A be an Abelian variety defined over K, with semi-stable reduction. In this note, we give a criterion for which $V_p{\left(A\right)}^{I_K}=0$, where $V_p\left(A\right)$ is the p-adic Tate module associated to A.