On the $L$-polynomials of curves over finite fields

TL;DR

This paper investigates the distribution of $L$-polynomial coefficients of algebraic curves over finite fields, establishing a dimension result for their span and proposing a conjecture on point distributions.

Contribution

It proves that the span of $L$-polynomials of genus $g$ curves over finite fields has dimension $g+1$, and introduces a conjecture on their Archimedean distribution.

Findings

The $Q$-vector space spanned by $L$-polynomials has dimension $g+1$.

Distribution results for coefficients of $L$-polynomials in a non-Archimedean setting.

A conjecture on the Archimedean distribution of rational points.

Abstract

We discuss, in a non-Archimedean setting, the distribution of the coefficients of $L$-polynomials of curves of genus $g$ over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such characteristic polynomials has dimension $g+1$. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.