  We discuss, in a non-archimedean setting, the distribution of the
coefficients of L-polynomials of curves of genus $g$ over $\mathbb{F}_q$. This
allows us to prove, among other things, 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.
