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Ramification bounds via Wach modules and q-crystalline cohomology

Pavel Čoupek

Abstract

Let $K$ be an absolutely unramified $p$-adic field. We establish a ramification bound, depending only on the given prime $p$ and an integer $i$, for mod $p$ Galois representations associated with Wach modules of height at most $i$. Using an instance of $q$-crystalline cohomology (in its prismatic form), we thus obtain improved bounds on the ramification of $\mathrm{H}^{i}_{et}(X_{\mathbb{C}_K}, \mathbb{Z}/p\mathbb{Z})$ for a smooth proper $p$-adic formal scheme $X$ over $\mathcal{O}_K$, for arbitrarily large degree $i$.

Ramification bounds via Wach modules and q-crystalline cohomology

Abstract

Let be an absolutely unramified -adic field. We establish a ramification bound, depending only on the given prime and an integer , for mod Galois representations associated with Wach modules of height at most . Using an instance of -crystalline cohomology (in its prismatic form), we thus obtain improved bounds on the ramification of for a smooth proper -adic formal scheme over , for arbitrarily large degree .

Paper Structure

This paper contains 6 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prisms $\mathbb{A}$ and $A_{\mathrm{inf}}$
  4. Ramification groups and Fontaine's property $(P_m)$
  5. Wach cohomology and Wach modules
  6. Ramification bound

Key Result

Theorem 1.1

Assume that $K$ is absolutely unramified. Let $T$ be a mod $p$ crystalline representation in the sense of (TorsionCrysAbstract) or (TorsionCrysGeometric) above, relative to the integer $i$. Then $G_K^{v}$ acts trivially on $T$ when where $\alpha$ is the least integer satisfying $p^{\alpha}>ip/(p-1)$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2: FontaineYoshidaCarusoLiu
  • Lemma 2.3: CarusoLiu
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 28 more