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

Abstract

Let C be a smooth irreducible projective curve defined over a finite field $\mathbb{F}_{q}$ of q elements of characteristic p>3 and $K=\mathbb{F}_{q}(C)$ its function field and $\phi_{\mathcal{E}}:\mathcal{E}\to C$ the minimal regular model of $\mathbf{E}/K$. For each $P\in C$ denote $\mathcal{E}_P=\phi^{-1}_{\mathcal{E}}(P)$. The elliptic curve $E/K$ has good reduction at $P\in C$ if and only if $\mathcal{E}_P$ is an elliptic curve defined over the residue field $\kappa_P$ of $P$. This field is a finite extension of $\mathbb{F}_q$ of degree $\deg(P)$. Let $t(\mathcal{E}_P)=q^{\deg(P)}+1-#\mathcal{E}_P(\kappa_P)$ be the trace of Frobenius at P. By Hasse-Weil's theorem (cf. [10, Chapter V, Theorem 2.4]), $t(\mathcal{E}_P)$ is the sum of the inverses of the zeros of the zeta function of $\mathcal{E}_P$. In particular, $|t(\mathcal{E}_P)|\le 2q^{\deg(P)}$. Let $C_0\subset C$ be the set of points of C at which $E/K$ has good reduction and $C_0(\mathbb{F}_{q^k})$ the subset of $\mathbb{F}_{q^k}$-rational points of $C_0$. We discuss the following question. Let $k\ge 1$ and t be integers and suppose $|t|\le 2q^{k/2}$. Let $\pi(k,t)=#\{P\in C_0(\mathbb{F}_{q^k}) | t(\mathcal{E}_P)=t\}$. How big is $\pi(k,t)$?