Bounds for the distribution of the Frobenius traces associated to a generic abelian variety

TL;DR

This paper establishes bounds on the distribution of Frobenius traces for generic abelian varieties over Q, under GRH and additional conjectures, revealing how often traces take specific values and their size relative to primes.

Contribution

It provides new explicit bounds on Frobenius trace distributions for generic abelian varieties assuming GRH and other conjectures, extending previous results.

Findings

Bounds on the number of primes with specific Frobenius traces under GRH.

Almost all primes satisfy a lower bound on the size of Frobenius traces.

Stronger bounds are obtained assuming additional conjectures like Artin's Holomorphy and Pair Correlation.

Abstract

Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, $A$ has a surjective residual modulo $\ell$ Galois representation. For $t\in \mathbb{Z}$ and $x>0$, denote by $\pi_A(x, t)$ the number of primes $p \leq x$ for which the Frobenius trace $a_{1, p}(A)$ associated to $A \pmod p$ equals $t$. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), we obtain that $\pi_A(x, 0) \ll_A x^{1 - \frac{1}{2g^2+g+1}}/(\log x)^{1 - \frac{2}{2g^2+g+1}}$ and $\pi_A(x, t) \ll_A x^{1 - \frac{1}{2g^2+g+2}}/(\log x)^{1 - \frac{2}{2g^2+g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)| > p^{\frac{1}{2 g^2 + g + 1}}/ (\log p)^{\frac{2}{2g^2+g+1}+\varepsilon}$ for any $\varepsilon>0$. Assuming, in addition to GRH, Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we obtain that $\pi_A(x, 0) \ll_A x^{1 - \frac{1}{g+1}}/(\log x)^{1 - \frac{4}{g+1}}$ and $\pi_A(x, t) \ll_A x^{1 - \frac{1}{g+2}}/(\log x)^{1 - \frac{4}{g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)|> p^{\frac{1}{g + 2} - \varepsilon }$ for any $\varepsilon>0$.