Distributions of Traces of Frobenius for Smooth Plane Curves Over Finite Fields

ABSTRACT

We analyze the variation around the mean of the distribution of the number of rational points on non-hyperelliptic genus 3 curves over finite fields, 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. We put our results in perspective with a numerical study for prime fields of characteristic 11 ⩽ p ⩽ 53. Our methods shed some new light on the asymmetry of the distribution around its mean value, which is related to the Serre obstruction.

KEYWORDS:

• genus 3 curves
• plane quartics
• moduli
• families
• enumeration
• finite fields

2000 AMS Subject Classification:

• 14Q05
• 14H10
• 14H25
• 14H37
• 14H45
• 14H50

Acknowledgments

We are very grateful to Mohamed Barakat for his helpful comments, Masaaki Homma for his proof of Lemma 3.2, Everett Howe for email exchanges which led us to the heuristic interpretation described in this article, and Atilla Yilmaz for pointing out the link between our statistical result and Edgeworth series.

Funding

The authors acknowledge support by grant ANR-09-BLAN-0020-01.