Arithmetic Properties of the Frobenius Traces Defined by a Rational Abelian Variety (with two appendices by J-P. Serre). (27th June 2016) - Cojocaru, Alina Carmen; Davis, Rachel; Silverberg, Alice; Stange, Katherine E.
Abstract: 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 $\rho_A$ . In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erdös-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 Lang and Trotter from 1976 about elliptic curves.