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