Adapted bases of Kisin modules and Serre weights

TL;DR

This paper investigates Kisin modules for crystalline Galois representations over tamely ramified extensions, establishing adapted bases under small Hodge-Tate weights and applying these to advance Serre's conjecture in specific cases.

Contribution

It introduces the concept of adapted bases for Kisin modules with small Hodge-Tate weights and applies this to prove new cases of the weight part of Serre's conjecture for $e=2$.

Findings

Existence of adapted bases for certain Kisin modules

Results on reductions and liftings of crystalline representations when $e=2$

New cases of the weight part of Serre's conjecture

Abstract

Let $p>2$ be a prime. Let $K$ be a tamely ramified finite extension over $\mathbb Q_p$ with ramification index $e$, and let $G_K$ be the Galois group. We study Kisin modules attached to crystalline representations of $G_K$ whose labeled Hodge-Tate weights are relatively small (a sort of "$er\le p$" condition where $r$ is the maximal Hodge-Tate weight). In particular, we show that these Kisin modules admit "adapted bases". We then apply these results in the special case $e=2$ to study reductions and liftings of certain crystalline representations. As a consequence, we establish some new cases of weight part of Serre's conjectures (when $e=2$).