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.

arXiv commentv4: Accepted version, to appear in Proceedings of the Royal Society of Edinburgh. At the suggestion of a referee we removed the appendix, which is still available as part of v3 of this paper here on the ArXiv

References in corpus (8)

• Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
• Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields
• Surjectivity of Galois Representations in Rational Families of Abelian Varieties
• The probability that the number of points on the Jacobian of a genus 2 curve is prime
• Elliptic curves, random matrices and orbital integrals
• Curves in characteristic 2 with non-trivial 2-torsion
• Crystalline Chebotarëv density theorems
• Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields