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

TL;DR

This paper studies the distribution and arithmetic properties of Frobenius trace values associated with rational abelian varieties, providing bounds, probabilistic results, and conjectures that extend classical results from elliptic curves to higher dimensions.

Contribution

It introduces new bounds and probabilistic theorems for Frobenius trace distributions of abelian varieties, generalizing classical elliptic curve conjectures.

Findings

Upper bounds for the count of primes with fixed Frobenius trace

An Erd"os-Kac type theorem for the number of prime factors of traces

A conjecture on the asymptotic distribution of Frobenius traces

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.