Bounds for the distribution of the Frobenius traces associated to a generic abelian variety

Abstract

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 $\ddot{I}{€}_A(x,t)$ the number of primes $p\le x$ for which the Frobenius trace $a_{1,p}(A)$ associated to $A(\mathrm{mod}p)$ equals $t$. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), we obtain that $\ddot{I}{€}_A(x,0){\ll }_A{x}^{1-\frac{1}{2g^2+g+1}}/(\log x{)}^{1-\frac{2}{2g^2+g+1}}$ and $\ddot{I}{€}_A(x,t){\ll }_A{x}^{1-\frac{1}{2g^2+g+2}}/(\log x{)}^{1-\frac{2}{2g^2+g+2}}$ if $t\ne 0$, and deduce that almost all primes $p$ satisfy $|a_{1,p}(A)|>p^{\frac{1}{2g^2+g+1}}/(\log p{)}^{\frac{2}{2g^2+g+1}+\epsilon }$ for any $\epsilon >0$. Assuming, in addition to GRH, Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we obtain that $\ddot{I}{€}_A(x,0){\ll }_A{x}^{1-\frac{1}{g+1}}/(\log x{)}^{1-\frac{4}{g+1}}$ and $\ddot{I}{€}_A(x,t){\ll }_A{x}^{1-\frac{1}{g+2}}/(\log x{)}^{1-\frac{4}{g+2}}$ if $t\ne 0$, and deduce that almost all primes $p$ satisfy $|a_{1,p}(A)|>p^{\frac{1}{g+2}-\epsilon }$ for any $\epsilon >0$.

41 pages, accepted by Mathematische Zeitschrift