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arXiv · 1504.00902

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

Alina Carmen Cojocaru·Rachel Davis·Alice Silverberg·Katherine E. Stange
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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 $\rho_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erd\"os-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 S. Lang and H. Trotter from 1976 about elliptic curves.

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Alina Carmen Cojocaru, Rachel Davis, Alice Silverberg, Katherine E. Stange. 2015-04-03. Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre). https://arxiv.org/abs/1504.00902

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