Crystalline lifts of semisimple $G$-valued Galois representations with fixed determinant

TL;DR

This paper proves the existence of crystalline lifts of semisimple Galois representations with fixed abelianization and regular Hodge-Tate weights, extending previous results to more general groups and parameters.

Contribution

It generalizes prior work by establishing crystalline lift existence for Galois representations valued in split reductive groups and their L-groups, with fixed abelianization.

Findings

Existence of crystalline lifts with prescribed abelianization.

Extension of results to quasi-split tame groups.

Generalization to L-parameters for semisimple mod p representations.

Abstract

For a finite extension $K/\mathbb{Q}_p$ and a split reductive group $G$ over $\mathcal{O}_K$, let $\overline{\rho} \colon \mathrm{Gal}_K \to G(\overline{\mathbb{F}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group $\mathrm{Gal}_K$. Let $\overline{\rho}^{\mathrm{ab}}$ be the abelianization of $\overline{\rho}$ and fix a crystalline lift $\psi$ of $\overline{\rho}^{\mathrm{ab}}$. We show the existence of a crystalline lift $\rho$ of $\overline{\rho}$ with regular Hodge-Tate weights such that the abelianization of $\rho$ coincides with $\psi$. We also show analogous results in the case that $G$ is a quasi-split tame group and $\overline{\rho} \colon \mathrm{Gal}_K \to {^L}G(\overline{\mathbb{F}}_p)$ is a semisimple mod $p$ $L$-parameter. These theorems are generalizations of those of Lin and B\"ockle-Iyengar-Pa\v{s}k\={u}nas.