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Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such thatthe image of its associated absolute Galois representation $\rho_A$ is open in$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic ofthe traces $a_{1, p}$ of the Frobenius at $p$ in$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under $\rho_A$, modulovarying primes $p$. In particular, we obtain upper bounds for the countingfunction $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erd\"os-Kac typetheorem for the number of prime factors of $a_{1, p}$. We also formulate aconjecture 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 1976about elliptic curves.

Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre)