  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
$\varepsilon>0$.
