Let $A$ be an abelian variety defined over a number field $K$. Let $v$ be a place of $K$ and denote by $K_v$ the $v$-adic completion of $K$ with respect to $||\cdot||_v$.
Assume $A$ is simple, is it true that $A\otimes K_v$ is also simple?
What if instead I assume that $A$ is geometrically simple?
