Crystalline lifts and a variant of the Steinberg-Winter theorem

TL;DR

This paper constructs crystalline lifts for mod p Galois representations over p-adic fields, explores rationality issues, and proves a variant of the Steinberg-Winter theorem, advancing understanding of p-adic Hodge theory and Galois representations.

Contribution

It provides a general method for constructing crystalline lifts of mod p Galois representations valued in reductive groups, and introduces a new variant of the Steinberg-Winter theorem.

Findings

Crystalline lifts are constructed for all irreducible mod p Galois representations.

The lifts are Hodge-Tate regular, satisfying specific p-adic Hodge theoretic conditions.

A new variant of the Steinberg-Winter theorem is established.

Abstract

Let $K/\mathbb{Q}_p$ be a finite extension. For all irreducible representations $\bar\rho: G_K \to G(\bar{\mathbb{F}}_p)$ valued in a general reductive group $G$, we construct crystalline lifts of $\bar\rho$ which are Hodge-Tate regular. We also discuss rationality questions. We prove a variant of the Steinberg-Winter theorem along the way.