Let $A$ be an abelian variety over $mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $rho_A$ is open in $operatorname{GSp}_{2g}(hat{mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ o
f the Frobenius at $p$ in $operatorname{Gal}(overline{mathbb{Q}}/mathbb{Q})$ under $rho_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $#{p leq x: a_{1, p} = t}$ and we prove an Erdos-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymptotic behaviour of $#{p leq x: a_{1, p} = t}$, which generalizes a well-known conjecture of S. Lang and H. Trotter from 1976 about elliptic curves.
