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 Q of dimension g such that the image of its associated absolute Galois representation ρ_A is open in GSp_2g( Z) . We investigate the arithmetic of the traces a_1, p of the Frobenius at p in Gal(Q/Q) under ρ_A . In particular, we obtain upper bounds for the counting function #{p ≤ 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 ≤ x: a_1, p = t}, which generalizes a well-known conjecture of Lang and Trotter from 1976 about elliptic curves.