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

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 ${\mathrm{GSp}}_{2g}(\widehat{\mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1,p}$ of the Frobenius at $p$ in $\mathrm{Gal}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})$ under ${\rho }_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\mathrm{\#}\{p\le x:a_{1,p}=t\}$ 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 $\mathrm{\#}\{p\le x:a_{1,p}=t\}$, which generalizes a well-known conjecture of S. Lang and H. Trotter from 1976 about elliptic curves.