  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 $a_{1, p}$ of 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 Erd\"os-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.
