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

Abstract

Let $g\ge 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to %the product $E_1\times \dots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\dots$, $E_g/\mathbb{Q}$, 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 $\ddot{I}{€}_A(x,t)$ the number of primes $p\le x$, 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 $\ddot{I}{€}_A(x,0){\ll }_A{x}^{1-\frac{1}{3g+1}}/(\log x{)}^{1-\frac{2}{3g+1}}$ and $\ddot{I}{€}_A(x,t){\ll }_A{x}^{1-\frac{1}{3g+2}}/(\log x{)}^{1-\frac{2}{3g+2}}$ if $t\ne 0$. 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 $|a_{1,p}(A)|>p^{\frac{1}{3g+1}-\epsilon }$ for any fixed $\epsilon >0$.