Distribution of primes of split reductions for abelian surfaces

TL;DR

This paper provides improved bounds on the distribution of primes where abelian surface reductions split, under GRH, depending on the endomorphism structure, advancing previous results by Achter and Zywina.

Contribution

It establishes new, sharper bounds for the count of split primes of abelian surfaces with various endomorphism rings, assuming GRH and other conjectures.

Findings

Bounds depend on endomorphism ring type (trivial, real multiplication, complex multiplication).

Results improve previous bounds by Achter (2012) and Zywina (2014).

Provides conditional bounds under GRH and other conjectures.

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.