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The local lifting problem for $(\mathbb{Z}/2\mathbb{Z})^3$

Guillaume Pagot

Abstract

Let $k$ be an algebraically closed field of characteristic $2$. In this paper we describe the $(\mathbb{Z}/2\mathbb{Z})^3$-actions on $k[[z]]$ for which there is a discrete valuation ring $R$, a finite extension of the ring of Witt vectors $W(k)$, such that they can be lifted as a group of $R$-automorphisms of $R[[Z]]$. In fact the necessary and sufficient condition for such an action to lift involves only the conductor type of the corresponding extension.

The local lifting problem for $(\mathbb{Z}/2\mathbb{Z})^3$

Abstract

Let be an algebraically closed field of characteristic . In this paper we describe the -actions on for which there is a discrete valuation ring , a finite extension of the ring of Witt vectors , such that they can be lifted as a group of -automorphisms of . In fact the necessary and sufficient condition for such an action to lift involves only the conductor type of the corresponding extension.
Paper Structure (20 sections, 11 theorems, 65 equations, 6 figures)

This paper contains 20 sections, 11 theorems, 65 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The (local) lifting problem for curves
  3. Tools for the proof
  4. The case $G=(\mathbb{Z} /2\mathbb{Z} )^2$
  5. Geometry of the branch locus
  6. Construction of covers with fixed geometry and reduction
  7. Aim
  8. Construction of the cover
  9. End of the proof of Theorem 1.a
  10. The case $(\mathbb{Z} /2\mathbb{Z} )^3$
  11. Geometry of the branch locus
  12. Choice of thicknesses
  13. Construction of covers with fixed geometry and reduction
  14. Aim
  15. Construction of the cover
  16. ...and 5 more sections

Key Result

theorem 1

Let $k$ an algebraically closed field of characteristic $2$. Let $n\in\mathbb{N} ^*$ (the set of positive integers). Let $G$ be a group of $k$-automorphisms of the ring $k[[z]]$ and $G\simeq (\mathbb{Z} /2\mathbb{Z} )^n$. We then have the following two results:

Figures (6)

  • Figure 1: Branch locus when $G=(\mathbb{Z} /2\mathbb{Z} )^2$
  • Figure 2: Branch loci of $Rev_1$ and $Rev_2$
  • Figure 3: Branch locus when $G=(\mathbb{Z} /2\mathbb{Z} )^3$
  • Figure 4: Branch loci of the three covers
  • Figure 5: Branch locus when $m_1=m_2=m_3$
  • ...and 1 more figures

Theorems & Definitions (12)

  • theorem 1
  • definition 1
  • theorem 2
  • lemma 1
  • lemma 2
  • Proposition 1
  • lemma 3
  • lemma 4
  • Proposition 2
  • Proposition 3
  • ...and 2 more