On the distribution of the of Frobenius elements on elliptic curves over function fields

Details On the distribution of the of Frobenius elements on elliptic curves over function fields On the distribution of the of Frobenius elements on elliptic curves over function fields

2002-11-20

arxiv.org/abs/math/0211315v1

Mathematics

Number Theory

8 pages

Scientific paper

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 $\alpha\le\theta_P\le\beta$.

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