Arithmetic Properties of the Frobenius Traces Defined by a Rational Abelian Variety (with two appendices by J-P. Serre). (27th June 2016)

• Record Type: Journal Article
• Authors: Cojocaru, Alina Carmen; Davis, Rachel; Silverberg, Alice; Stange, Katherine E.
• Is Part Of: International mathematics research notices. Volume 2017:Number 12(2017)
• Page Start: 3557
• Page End: 3602
• Publication Date: 2016-06-27
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw058
• Languages: English
• ISSNs: 1073-7928

Abstract

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.