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

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.