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

arXiv2011.08773MaRDI QIDQ6504286

Abstract: We show for all local fields

K / m a t h b b Q_{p}

, with

p > 3

, all representations

admit a crystalline lift

, where

G_{2}

is the exceptional Chevalley group of type

G_{2}

. We also show the existence of crystalline lifts of representations valued in classical groups assuming the existence of Emerton-Gee stack valued in general reductive groups and some codimension estimates on the Emerton-Gee stack. 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

(p h i, G a m m a)

-modules to contruct abelian extensions.

Mathematics Subject Classification ID

Galois representations (11F80)

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