Abelian varieties that split modulo all but finitely many primes

Abstract

Let $A$ be a simple abelian variety over a number field $k$ such that $\mathrm{End}(A)$ is noncommutative. We show that $A$ splits modulo all but finitely many primes of $k$. We prove this by considering the subalgebras of $\mathrm{End}(A_{\mathfrak{p}})\otimes \mathbb{Q}$ which have prime Schur index. Our main tools are Tate's characterization of endomorphism algebras of abelian varieties over finite fields, and a Theorem of Chia-Fu Yu on embeddings of simple algebras.

8 pages, comments are welcome!