Lyndon-Demushkin method and crystalline lifts of $G_2$-valued Galois representations
Zhongyipan Lin
TL;DR
This work proves that every mod $p$ Galois representation valued in the exceptional group ${G_2}$ over a $p$-adic field with $p>3$ admits a crystalline lift to ${G_2}$. The authors develop a fully explicit obstruction theory based on Lyndon–Demuškin cochains, replacing non-abelian obstructions with tractable abelian cup-product data via Heisenberg quotients, and they integrate Emerton–Gee stacks to control deformations. A central technical achievement is the construction and analysis of the Lyndon–Demuškin complex for both abelian and nilpotent coefficients, together with precise cup-product calculus modulo $\varpi$ and over residue fields. They then combine these local obstructions with the Emerton–Gee framework to produce crystalline lifts for ${G_2}$, including compatibility of lifts with chosen parabolic Levi factors, and they provide codimension estimates for the loci where obstructions persist. The results illustrate a broad strategy to extend crystalline lifting results beyond classical groups to exceptional groups by leveraging explicit combinatorial group theory and modern stacks techniques.
Abstract
We show for all local fields $K/\mathbb{Q}_p$, with $p >3$, all representations $\barρ:G_K \to G_2(\bar{\mathbb{F}}_p)$ admit a crystalline lift $ρ: G_K\to G_2(\bar{\mathbb{Z}}_p)$, where $G_2$ is the exceptional Chevalley group of type $G_2$. The main ingredient is a new technique for analyzing the obstruction of non-abelian extensions of Galois modules, which has roots in combinatorial group theory. We also rely on the Emerton-Gee stack of $(φ, Γ)$-modules to construct abelian extensions.