Distribution of primes of split reductions of abelian surfaces

In this talk, we focus on reductions of an absolutely simple abelian surface $A$ defined over a number field $K$. The Murty-Patankar Conjecture holds in this case, indicating that the density of primes of split reduction of $A$ is 0 if and only if the geometric endomorphism ring of $A$ is commutative. Motivated by the this conjecture and assuming $A$ has a commutative geometric endomorphism ring, we prove nontrivial upper bounds for the number of primes $\mathfrak{p}$ of $K$ with norm bounded by $x$, for which the reduction $A_{\mathfrak{p}}$ splits into a product of elliptic curves. These bounds improve prior results by Achter in 2012 and Zywina in 2014.