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.
Categories: math.NT
