Ramification estimate for Fontaine-Laffaille Galois modules

TL;DR

This paper provides a new proof of Fontaine's conjecture regarding the triviality of certain ramification subgroup actions on specific Galois modules, addressing gaps in previous proofs and extending the validity.

Contribution

The paper offers a novel proof of Fontaine's conjecture for torsion crystalline Galois modules with bounded Hodge-Tate weights, improving upon earlier incomplete proofs.

Findings

Proves Fontaine's conjecture for a broader class of Galois modules.

Addresses and corrects gaps in previous proofs.

Extends the understanding of ramification subgroup actions on crystalline modules.

Abstract

Suppose $K$ is unramified over $\mathbb Q _p$ and $\Gamma _K=\operatorname{Gal}(\bar K/K)$. Let $H$ be a torsion $\Gamma _K$-equivariant subquotient of crystalline $\mathbb Q _p[\Gamma _K]$-module with HT weights from $[0,p-2]$. We give a new proof of Fontaine's conjecture about the triviality of action of some ramification subgroups $\Gamma _K^{(v)}$ on $H$. The earlier author's proof from [1] contains a gap and proves this conjecture only for some subgroups of index $p$ in $\Gamma _K^{(v)}$.