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

TL;DR

This paper provides a heuristic explanation for the distribution of Frobenius traces in smooth plane curves over finite fields, highlighting asymmetry related to the Serre obstruction, based on extrapolated data and previous results.

Contribution

It introduces a new heuristic approach to understanding Frobenius trace distributions for plane curves, connecting asymmetry to the Serre obstruction.

Findings

Distribution of Frobenius traces shows asymmetry around the mean.

Heuristic explanations align with observed data for small degree curves.

Insights into the Serre obstruction's role in trace distribution asymmetry.

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.