Ramification bounds via Wach modules and q-crystalline cohomology

TL;DR

This paper derives new bounds on the ramification of certain Galois representations and étale cohomology groups over unramified p-adic fields, using Wach modules and q-crystalline cohomology techniques.

Contribution

It introduces a ramification bound depending only on p and i for mod p Galois representations from Wach modules, utilizing q-crystalline cohomology for improved ramification estimates.

Findings

Established ramification bounds for Galois representations from Wach modules.

Improved bounds on ramification of étale cohomology groups for p-adic formal schemes.

Applied q-crystalline cohomology to enhance understanding of ramification behavior.

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 $i$.