Distribution of supersingular primes for abelian surfaces

Let A/Q be an absolutely simple abelian surface. We give unconditional upper bounds for the number of primes p up to x such that A has supersingular reduction at p, in the cases where A has a trivial endomorphism ring, real multiplication, and quaternion multiplication, respectively. Particularly, in the real multiplication case, the bound is related to an upper bound for the distribution of Frobenius traces of A. Also in the above case, we provide unconditional upper bounds for a modified version of the problem, that concerns the number of primes for which the reduction of A at the prime has a particular Newton datum and is not simple.