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

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

2001-11-09

arxiv.org/abs/math/0111105v3

Acta Arithmetica, 106.3 (2003), 255-263

Mathematics

Number Theory

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

Scientific paper

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 $\pi(k,t)$?

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