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Arithmetic Properties of the Frobenius Traces Defined by a Rational Abelian Variety (with two appendices by J-P. Serre)
journal contribution
posted on 2018-06-28, 00:00 authored by AC Cojocaru, R Davis, A Silverberg, KE StangeLet 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.
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Publisher Statement
"This is a copy of an article published in the International Mathematics Research Notices © 2017 OXFORD UNIV PRESS"Publisher
Oxford University PressLanguage
- en_US
issn
1073-7928Issue date
2017-06-01Usage metrics
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