Let A be an abelian variety defined over \({\mathbb {Q}}\) and of dimension g. Assume that, for each sufficiently large prime \(\ell \) , A has a surjective residual modulo \(\ell \) Galois representation. For \(t\in {\mathbb {Z}}\) and \(x>0\) , denote by \(\pi _A(x, t)\) the number of primes \(p \le x\) for which the Frobenius trace \(a_{1, p}(A)\) associated to \(A (\text {mod}p)\) equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), we obtain that \(\pi _A(x, 0) \ll _A x^{1 - \frac{1}{2g^2+g+1}}/(\text {log}x)^{1 - \frac{2}{2g^2+g+1}}\) and \(\pi _A(x, t) \ll _A x^{1 - \frac{1}{2g^2+g+2}}/(\text {log}x)^{1 - \frac{2}{2g^2+g+2}}\) if \(t \ne 0\) , and deduce that almost all primes p satisfy \(|a_{1, p}(A)| > p^{\frac{1}{2 g^2 + g + 1}}/ (\text {log}p)^{\frac{2}{2g^2+g+1}+\varepsilon }\) for any \(\varepsilon >0\) . Assuming, in addition to GRH, Artin’s Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we obtain that \(\pi _A(x, 0) \ll _A x^{1 - \frac{1}{g+1}}/(\text {log}x)^{1 - \frac{4}{g+1}}\) and \(\pi _A(x, t) \ll _A x^{1 - \frac{1}{g+2}}/(\text {log}x)^{1 - \frac{4}{g+2}}\) if \(t \ne 0\) , and deduce that almost all primes p satisfy \(|a_{1, p}(A)|> p^{\frac{1}{g + 2} - \varepsilon }\) for any \(\varepsilon >0\) .
