Distribution of supersingular primes for abelian surfaces

Abstract

Let $A/K$ be an absolutely simple abelian surface defined over a number field $K$. We give unconditional upper bounds for the number of prime ideals $\mathfrak{p}$ of $K$ with norm up to $x$ such that $A$ has supersingular reduction at $\mathfrak{p}$. These bounds are obtained in three distinct settings, depending on the endomorphism algebra of $A$, namely, the case of trivial endomorphisms, real multiplication (RM), and quaternion multiplication (QM). In the RM case and when $K=\mathbb{Q}$, our results further implies an unconditional upper bound on the distribution of Frobenius traces of $A$. Furthermore, in the RM setting, we study the distribution of the middle coefficients of Frobenius polynomials of $A$ at primes where the reduction of $A$ splits.