Distribution of primes of split reductions for abelian surfaces

Abstract: Let $A$ be an absolutely simple abelian surface defined over a number field $K$ with a commutative (geometric) endomorphism ring. Let $\pi_{A, \text{split}}(x)$ denote the number of primes $\mathfrak{p}$ in $K$ such that each prime has norm bounded by $x$, of good reduction for $A$, and the reduction of $A$ at $\mathfrak{p}$ splits. It is known that the density of such primes is zero. Under the Generalized Riemann Hypothesis for Dedekind zeta functions and possibly extending the field $K$, we prove that $\pi_{A, \text{split}}(x) \ll_{A, K} x^{\frac{41}{42}}\log x$ if the endomorphism ring of $A$ is trivial; $\pi_{A, \text{split}}(x) \ll_{A, F, K} \frac{x^{\frac{11}{12}}}{(\log x)^{\frac{2}{3}}}$ if $A$ has real multiplication by a real quadratic field $F$; $\pi_{A, \text{split}}(x) \ll_{A, F, K} x^{\frac{2}{3}}(\log x)^{\frac{1}{3}}$ if $A$ has complex multiplication by a CM field $F$. These results improve the bounds by J. Achter in 2012 and D. Zywina in 2014. We also provide better bounds under other credible conjectures.