Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local Monodromy of 1-Dimensional p-Divisible Groups

Tristan Phillips

TL;DR

Let $R$ be a complete DVR of characteristic $p$ and $G$ a connected $1$-dimensional $p$-divisible group over $R$. The generic fiber yields a Galois representation $ ho$ whose image carries both a ramification filtration and a Lie filtration; the paper establishes an equicharacteristic analogue of Sen's theorem for this setting, relating the filtrations in the $1$-dimensional case and under an open-image hypothesis, extending Gross. It then proves that the slope-zero piece $ ho^{0/1}$ is irreducible, generalizing Chai, by analyzing the associated towers via the formal-group/Artin–Schreier framework. A key tool is the equivalence between connected $p$-divisible groups and formal A-modules, which allows explicit computations of ramification using Artin–Schreier theory and Newton polygons. The work culminates in detailed ramification information for the determinant and norm characters, linking the local monodromy to the arithmetic of the underlying formal and $p$-divisible structures and providing a fixed version of Cha00 under an open-image assumption.

Abstract

Let $G$ be a $p$-divisible group over a complete discrete valuation ring $R$ of characteristic $p$. The generic fiber of $G$ determines a Galois representation $ρ$. The image of $ρ$ admits a ramification filtration and a Lie filtration. We relate these filtrations in the case $G$ is one dimensional, giving an equicharacteristic version of Sen's theorem in this setting. This result generalizes a result of Gross. Additionally, we prove that the representation associated to the étale part of $G$ is irreducible, generalizing a result of Chai.

Local Monodromy of 1-Dimensional p-Divisible Groups

TL;DR

Let be a complete DVR of characteristic and a connected -dimensional -divisible group over . The generic fiber yields a Galois representation whose image carries both a ramification filtration and a Lie filtration; the paper establishes an equicharacteristic analogue of Sen's theorem for this setting, relating the filtrations in the -dimensional case and under an open-image hypothesis, extending Gross. It then proves that the slope-zero piece is irreducible, generalizing Chai, by analyzing the associated towers via the formal-group/Artin–Schreier framework. A key tool is the equivalence between connected -divisible groups and formal A-modules, which allows explicit computations of ramification using Artin–Schreier theory and Newton polygons. The work culminates in detailed ramification information for the determinant and norm characters, linking the local monodromy to the arithmetic of the underlying formal and -divisible structures and providing a fixed version of Cha00 under an open-image assumption.

Abstract

Let be a -divisible group over a complete discrete valuation ring of characteristic . The generic fiber of determines a Galois representation . The image of admits a ramification filtration and a Lie filtration. We relate these filtrations in the case is one dimensional, giving an equicharacteristic version of Sen's theorem in this setting. This result generalizes a result of Gross. Additionally, we prove that the representation associated to the étale part of is irreducible, generalizing a result of Chai.

Paper Structure

This paper contains 22 sections, 22 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. $p$-Adic Lie Groups
  4. Ramification Theory
  5. Lower ramification filtration
  6. Upper ramification filtration
  7. Newton Polygons
  8. Generalized Artin--Schreier Extensions
  9. Galois Representations of $p$-Divisible Groups in Characteristic $p$
  10. Formal Groups and Formal Modules
  11. Formal $A$-modules over rings of characteristic $p$
  12. $A$-typical formal modules
  13. From $p$-divisible groups to formal Modules
  14. Towers Arising From $\rho^{0/1}$
  15. The torsion tower and its subtowers
  16. ...and 7 more sections

Key Result

Theorem 2.0.2

DdMS99. A group $G$ is a $p$-adic Lie group if and only if it is a topological group containing an open uniform pro-$p$ subgroup.

Theorems & Definitions (54)

  • Definition 2.0.1: Uniform pro-$p$ group
  • Theorem 2.0.2
  • proof
  • Corollary 2.0.3
  • Theorem 2.0.4: Closed subgroup theorem
  • proof
  • Example 2.0.5
  • Proposition 3.1.1
  • Definition 3.1.2
  • Definition 3.2.1
  • ...and 44 more