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

Abstract

We show for all local fields $K/{\mathbb{Q}}_p$, with $p>3$, all representations $\bar{\rho }:G_K\to G_2({\bar{\mathbb{F}}}_p)$ admit a crystalline lift $\rho :G_K\to G_2({\bar{\mathbb{Z}}}_p)$, 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 $(\varphi ,\Gamma )$-modules to construct abelian extensions.

Accepted by ANT. Final version

References in corpus (1)

• G-valued local deformation rings and global lifts