Let $g \geq 1$ be an integer and let $A$ be an abelian variety defined over $\mathbb{ Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, the image of the residual modulo $\ell$ Galois representation of $A$ is isomorphic to $\text{ GSp}_{ 2g}(\mathbb{ Z}/\ell\mathbb{ Z})$. For an integer $t$ and a positive real number $x$, denote by $\pi_A(x, t)$ the number of primes $p \leq x$, of good reduction for $A$, for which the Frobenius trace $a_{ 1, p}(A)$ associated to the reduction of $A$ modulo $p$ equals $t$. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove 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 that $\pi_A(x, t) \ll_A x^{ 1-\frac{ 1}{ 2g^2+g+2}}/(\log x)^{ 1-\frac{ 2}{ 2g^2+g+2}}$ if $t \neq 0$. Under the assumptions stated above, we also prove the existence of a density one set of primes $p$ satisfying $|a_{ 1, p}(A)|> p^{ \frac{ 1}{ 2g^2+g+1}}/(\log p)^{ \varepsilon}$ for any fixed $\varepsilon>0$. Assuming, in addition to the Generalized Riemann Hypothesis for Dedekind zeta functions, Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we prove that $\pi_A(x, 0) \ll_A x^{ 1-\frac{ 1}{ g+1}}/(\log x)^{ 1-\frac{ 4}{ g+1}}$ and that $\pi_A(x, t) \ll_A x^{ 1-\frac{ 1}{ g+2}}/(\log x)^{ 1-\frac{ 4}{ g+2}}$ if $t \neq 0$, and we deduce the existence of a density one set of primes $p$ satisfying $|a_{ 1, p}(A)|> p^{ \frac{ 1}{ g+2}-\varepsilon}$ for any fixed $\varepsilon>0$. These are currently the best known conditional upper bounds for $\pi_A(x, t)$ and the best known conditional lower bounds for $|a_{ 1, p}(A)|$, for most primes $p$.
Categories: math.NT