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

TL;DR

The paper establishes the existence of crystalline lifts of quasi-semisimple $G$-valued mod $p$ Galois representations with fixed abelianization and regular Hodge–Tate weights, after a finite extension of the coefficient field; it also proves analogous potentially crystalline lifts for semisimple $L$-parameters in the quasi-split tame setting. The construction combines Lin’s method for crystalline lifts with fixed abelianizations (via inertial- and Frobenius-compatibility criteria) and BIP’s norm-control techniques, augmented by Lubin–Tate characters to achieve HT-regularity, and further extends these results to L-parameters through Langlands–Shelstad factorizations. By ensuring fixed abelianizations and HT-regular weights, these results advance the understanding of crystalline points on deformation spaces and have implications for the Zariski-density of crystalline lifts on Emerton–Gee stacks. The work unifies and extends prior results of Lin (for split $G$) and Böckle–Iyengar–Paškūnas (for fixed-determinant or abelianized settings) to the broader context of $G$-valued representations and their Langlands dual parameters.

Abstract

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

Theorems & Definitions (31)

• Theorem 1.1: Theorem \ref{['qss_cryslift_fixeddet']}
• Theorem 1.2: Theorem \ref{['qst_dRlift_fixeddet']}
• Remark 1.3
• Definition 3.1
• Theorem 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof
• Lemma 3.5