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Lifting Galois representations via Kummer flags

Andrea Conti, Cyril Demarche, Mathieu Florence

Abstract

Let $Γ$ be either i) the absolute Galois group of a local field $F$, or ii) the topological fundamental group of a closed connected orientable surface of genus $g$. In case i), assume that $μ_{p^2} \subset F$. We give an elementary and unified proof that every representation $ρ_1: Γ\to \mathbf{GL}_d(\mathbb{F}_p)$ lifts to a representation $ρ_2: Γ\to \mathbf{GL}_d(\mathbb{Z}/p^2)$. [In case i), it is understood these are continuous.] The actual statement is much stronger: for all $r \geq 1$, under "suitable" assumptions, triangular representations $ρ_r: Γ\to \mathbf{B}_d(\mathbb{Z}/p^r)$ lift to $ρ_{r+1}: Γ\to \mathbf{B}_d(\mathbb{Z}/p^{r+1})$, in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of $\textit{Kummer flag}$. An essential aspect of this work is to identify the common properties of groups i) and ii) that suffice to ensure the existence of such lifts.

Lifting Galois representations via Kummer flags

Abstract

Let be either i) the absolute Galois group of a local field , or ii) the topological fundamental group of a closed connected orientable surface of genus . In case i), assume that . We give an elementary and unified proof that every representation lifts to a representation . [In case i), it is understood these are continuous.] The actual statement is much stronger: for all , under "suitable" assumptions, triangular representations lift to , in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of . An essential aspect of this work is to identify the common properties of groups i) and ii) that suffice to ensure the existence of such lifts.
Paper Structure (12 sections, 21 theorems, 139 equations)

This paper contains 12 sections, 21 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Flags
  4. Teichmüller lift
  5. Galois cohomology and $p$-manageable groups
  6. Common properties shared by (seemingly) unrelated groups
  7. When $H^1(\Gamma,\mathbb{F}_p)$ is finite
  8. Gluing, lifting and gluifting extensions of $(\Gamma,r)$-bundles
  9. Lifting (wound) Kummer flags
  10. Wound Kummer flags
  11. Generalisation to a larger class of coefficients
  12. Kummer flags, in the presence of enough roots of unity

Key Result

Theorem 1

Let $d,r \geq 1$ be two integers, and let $\rho_r\colon\Gamma\longrightarrow\mathbf{GL}_d(\mathbb{Z}/p^r)$ be a continuous representation equipped with a Kummer flag. In the arithmetic case, assume that $F$ contains a primitive $p^{r+1}$-th root of unity. Then $\rho_r$ admits a lift $\rho_{r+1}\colo

Theorems & Definitions (66)

  • Theorem
  • Corollary
  • Corollary
  • Corollary
  • Remark 1.1
  • Remark
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 56 more