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

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 $\varepsilon>0$.