Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A census of genus 6 curves over $\mathbb{F}_2$

Yongyuan Huang, Kiran S. Kedlaya, Jun Bo Lau

Abstract

We compile a complete list of isomorphism class representatives of curves of genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_6$, due to Mukai in the generic case. Our computed value of $\#\mathcal{M}_6(\mathbb{F}_2)$ agrees with the Lefschetz trace formula as recently computed by Bergstrom--Canning--Petersen--Schmitt.

A census of genus 6 curves over $\mathbb{F}_2$

Abstract

We compile a complete list of isomorphism class representatives of curves of genus 6 over . We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space , due to Mukai in the generic case. Our computed value of agrees with the Lefschetz trace formula as recently computed by Bergstrom--Canning--Petersen--Schmitt.
Paper Structure (14 sections, 7 theorems, 16 equations, 1 table)

This paper contains 14 sections, 7 theorems, 16 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Brill--Noether stratification of M6
  3. Tabulation of data
  4. Hyperelliptic curves
  5. Bielliptic curves
  6. Smooth plane quintic curves
  7. Trigonal curves of Maroni invariant 0
  8. Trigonal curves of Maroni invariant 2
  9. Generic curves
  10. Postprocessing
  11. Consistency checks
  12. Point counting on M6
  13. Point counts with marked points
  14. Point counts in strata

Key Result

Theorem 1.1

We obtain an explicit list of isomorphism class representatives for $\mathcal{M}_6(\mathbb{F}_2)$: it consists of $72227$ elements, and

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • ...and 5 more