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The Classification of the Stable Marked Reduction of Genus 2 Curves in Residue Characteristic 2

Tim Gehrunger

Abstract

Consider a hyperelliptic curve of genus $2$ over a field $K$ of characteristic zero. After extending $K$ we can view it as a marked curve with its $6$ Weierstrass points. We classify the structure of the potentially stable reduction of such curves for a valuation of residue characteristic $2$. We implement this classification into a computer algebra system and compute it for a list of curves defined over $\mathbb{Q}$ with conductor at most $2^{20}$.

The Classification of the Stable Marked Reduction of Genus 2 Curves in Residue Characteristic 2

Abstract

Consider a hyperelliptic curve of genus over a field of characteristic zero. After extending we can view it as a marked curve with its Weierstrass points. We classify the structure of the potentially stable reduction of such curves for a valuation of residue characteristic . We implement this classification into a computer algebra system and compute it for a list of curves defined over with conductor at most .

Paper Structure

This paper contains 11 sections, 4 theorems, 7 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Set-up
  3. Models of curves
  4. Hyperelliptic curves
  5. Thickness bound
  6. A classification for genus 2 curves
  7. Classification with thickness
  8. Computing $\delta'$
  9. Examples
  10. Implementation
  11. Curves with small conductor

Key Result

Theorem 1

There are 54 cases for the combinatorial structure of $C_0$. The space of parameters $(\alpha, \beta, \gamma, \delta', \varepsilon)$ decomposes into half-open polyhedral regionsHere we mean regions in Euclidean space described by a finite set of linear equalities and strict linear inequalities.$P_i\

Figures (16)

  • Figure 1: The possibilities for $(\bar{C}_0,\bar{p}_1,\ldots,\bar{p}_6)$ in genus $2$.
  • Figure 2: The possibilities for $(C_0,p_1,\ldots,p_6)$ in the case (A).
  • Figure 3: The possibilities for $(C_0,p_1,\ldots,p_6)$ in the case (B).
  • Figure 4: The possibilities for $(C_0,p_1,\ldots,p_6)$ in the case (C).
  • Figure 5: The first 12 possibilities for $(C_0,p_1,\ldots,p_6)$ in the case (D).
  • ...and 11 more figures

Theorems & Definitions (9)

  • Theorem 1: See Theorem \ref{['Genus2SummaryTheorem']} and Theorem \ref{['DeltaPrimeTheorem']}
  • Conjecture 2: = Conjecture \ref{['RealisationConjecture']}
  • Remark 2.1.1
  • Proposition 2.1.2
  • Definition 2.2.2
  • Theorem 3.1.2
  • Theorem 3.2.1
  • Example 4.1.1
  • Conjecture 4.2.1