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

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 $K∕{\mathbb{Q}}_p$, with $p>3$, all representations $\overline{\rho }:G_K\to G_2({\overline{\mathbb{𝔽}}}_p)$ admit a crystalline lift $\rho :G_K\to G_2({\overline{\mathbb{Z}}}_p)$, where $G_2$ is the exceptional Chevalley group of type $G_2$.