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

2015/04/03 by Cojocaru, Alina Carmen, Davis, Rachel, Silverberg, Alice +1
#11G10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1504.00902

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, modulo varying primes p. 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 S. Lang and H. Trotter from 1976 about elliptic curves.

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