The Lyndon–Demushkin method and crystalline lifts of G2-valued Galois representations

Lin, Zhongyipan

doi:10.2140/ant.2025.19.415

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

The Lyndon–Demushkin method and crystalline lifts of $G_2$-valued Galois representations

Zhongyipan Lin

Vol. 19 (2025), No. 3, 415–456

DOI: 10.2140/ant.2025.19.415

Abstract

We develop obstruction theory for lifting characteristic- $p$ local Galois representations valued in reductive groups of type $B_{l}$ , $C_{l}$ , $D_{l}$ or $G_{2}$ . An application of the Emerton–Gee stack then reduces the existence of crystalline lifts to a purely combinatorial problem when $p$ is not too small.

As a toy example, we show for all local fields $K ∕ ℚ_{p}$ , with $p > 3$ , all representations $\bar{ρ} : G_{K} \to G_{2} ({\bar{𝔽}}_{p})$ admit a crystalline lift $ρ : G_{K} \to G_{2} ({\bar{ℤ}}_{p})$ , where $G_{2}$ is the exceptional Chevalley group of type $G_{2}$ .

Keywords

Galois representations, exceptional group

Mathematical Subject Classification

Primary: 11F80

Milestones

Received: 25 August 2021

Revised: 25 December 2023

Accepted: 5 March 2024

Published: 20 February 2025

Authors

Zhongyipan Lin
Northwestern University Evanston, IL United States

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 19, No. 3, 2025