Distribution of the traces of Frobenius on elliptic curves over function fields

Abstract

Let $C$ be a smooth projective curve over $\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $\phi:\mathcal{E}\to C$ its minimal regular model. For each $P\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\mathcal{E}_P=\phi^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\mathcal{E}_P$ over the residue field $\kappa_P$ of $P$ are of the form $q_P^{1/2}e^{i\theta_P},q_{P}e^{-i\theta_P}$, where $q_P=q^{\deg(P)}$ and $0\le\theta_P\le\pi$. The goal of this note is to determine given an integer $B\ge 1$, $\alpha,\beta\in[0,\pi]$ the number of $P\in C$ where the reduction of $E$ is good and such that $\deg(P)\le B$ and $\alpha\le\theta_P\le\beta$.