Exceptional splitting of abelian surfaces over global function fields. 

Let $A$ denote a non-constant ordinary abelian surface over a global function field (of characteristic p) with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. Then we prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves. This is joint work with Davesh Maulik and Yunqing Tang.