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 Q of dimension g such that the image of its associated absolute Galois representation rho(A) is open in GSp(2g)((Z) over cap). We investigate the arithmetic of the traces a(1,p) of the Frobenius at p in Gal((Q) over bar /Q) under rho(A). In particular, we obtain upper bounds for the counting function #{p <= x : a(1,p) = t} and we prove an Erdos-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.