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On invariants of Artin-Schreier curves

Juanita Duque-Rosero, Heidi Goodson, Elisa Lorenzo García, Beth Malmskog, Renate Scheidler

Abstract

The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and determine all isomorphisms between curves in this form. We then compute reconstructing systems of invariants for curves in each connected component of the strata of the moduli spaces for Artin-Schreier curves of genus 3 and 4 for $p>2$.

On invariants of Artin-Schreier curves

Abstract

The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and determine all isomorphisms between curves in this form. We then compute reconstructing systems of invariants for curves in each connected component of the strata of the moduli spaces for Artin-Schreier curves of genus 3 and 4 for .
Paper Structure (21 sections, 37 theorems, 65 equations, 3 tables)

This paper contains 21 sections, 37 theorems, 65 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. A brief introduction to invariant theory
  4. Artin-Schreier curves
  5. Fractional linear transformations acting on FFp(x)
  6. Standard form
  7. Isomorphisms Between Curves in Standard Form
  8. Standard forms of exceptional curves
  9. Invariant Computations
  10. Genus 3, characteristic 3
  11. One pole of order 4
  12. One pole of order 2 and one pole of order 1
  13. Genus 3, characteristic 7
  14. Genus 4, characteristic 3
  15. A single pole of order 5
  16. ...and 6 more sections

Key Result

Theorem 1

A system of reconstructing invariants for all Artin-Schreier curves of genus $g=3,4$ in characteristic $p>2$ is given in Table tab:invariants.

Theorems & Definitions (75)

  • Theorem 1
  • Definition 2.1
  • Proposition 2.2: Noether Normalization Lemma
  • Theorem 2.3: Hilbert Finiteness Theorem
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Molien-Weyl Formula, DK02
  • Corollary 2.7
  • Lemma 2.8: Noether, Fleischmann, Benson, Fogarty, DK02
  • Definition 2.9
  • ...and 65 more