# Semisimplicity of the p-adic étale Tate module over $F_p(t)$

Let $k$ be a finitely generated field of positive characteristic p. Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural action of $Gal(k^{sep}|k)$ over $T_p(A)$ is semisimple?

• mathoverflow.net/questions/275355/… – user19475 Jun 29 '18 at 10:46
• @TKe thank you for your answer. In that answer, k is a finite field, while here k is a finitely generated field. For the moment i'm not able to deduce the case when k is finitely generated from the situation when k is finite. – Emiliano Ambrosi Jun 29 '18 at 15:46