Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

Abstract

Let $g \geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to %the product $E_1 \times \ldots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\ldots$, $E_g/\mathbb{Q}$, without complex multiplication and pairwise non-isogenous over $\overline{\mathbb{Q}}$. a product of $g$ elliptic curves defined over $\mathbb{Q}$, pairwise non-isogenous over $\overline{\mathbb{Q}}$ and each without complex multiplication. %pairwise non-isogenous over $\overline{\mathbb{Q}}$. 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 %the abelian variety $A$, 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 $\pi_A(x, 0) \ll_A x^{1 - \frac{1}{3 g+1 }}/(\log x)^{1 - \frac{2}{3 g+1}}$ and $\pi_A(x, t) \ll_A x^{1 - \frac{1}{3 g + 2}}/(\log x)^{1 - \frac{2}{3 g + 2}}$ if $t \neq 0$. These bounds largely improve upon recent ones obtained for $g = 2$ by H. Chen, N. Jones, and V. Serban, and may be viewed as generalizations to arbitrary $g$ of the bounds obtained for $g=1$ by M.R. Murty, V.K. Murty, and N. Saradha, combined with a refinement in the power of $\log x$ by D. Zywina. Under the same assumptions, we also prove the existence of a density one set of primes $p$ satisfying $|a_{1, p}(A)|>p^{\frac{1}{3 g + 1} - \varepsilon}$ for any fixed $\varepsilon>0$.