Distributions of traces of Frobenius for smooth plane curves over finite fields

Abstract

In a previous article, we obtained data on the distribution of traces of Frobenius of non-hyperelliptic genus $3$ curves over small finite fields. In the present one, we give a heuristic explanation of these data, by extrapolating from results on the distribution of traces of Frobenius for plane curves whose degree is small with respect to the cardinality of their finite base field. In particular, our methods shed some new light on the asymmetry of the distribution around its mean value, which is related to the Serre obstruction.

arXiv comment12 pages, 4 figures

References in corpus (9)

• Bertini theorems over finite fields
• Statistics for traces of cyclic trigonal curves over finite fields
• The fluctuations in the number of points of smooth plane curves over finite fields
• Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields
• Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves
• Distribution of zeta zeroes of Artin--Schreier curves
• A heuristic for the distribution of point counts for random curves over a finite field
• Courbes de genre 3 avec S3 comme groupe d'automorphismes
• The second largest number of points of plane curves over finite fields

Cited by in corpus (2)

• Most plane curves over finite fields are not blocking
• Refinements of Katz-Sarnak theory for the number of points on curves over finite fields