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 ${\varphi }_{\mathcal{E}}:\mathcal{E}\to C$ the minimal regular model of $\mathbf{E}/K$. For each $P\in C$ denote ${\mathcal{E}}_P={\varphi }_{\mathcal{E}}^{-1}(P)$. 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 $°(P)$. Let $t(\mathcal{E}_P)=q^{°(P)}+1-#\mathcal{E}_P(κ_P)$ 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, $|t({\mathcal{E}}_P)|\le 2q^{°(P)}$. 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 $π(k,t)=#\{P\in C_0(\mathbb{F}_{q^k}) | t(\mathcal{E}_P)=t\}$. How big is $\pi (k,t)$?

11 pages, replaced version, minor correction on the degree of the j-map