Combinatorics of Serre weights in the potentially Barsotti-Tate setting

TL;DR

This paper explores the relationship between Kisin varieties and Serre weights in the context of potentially Barsotti--Tate Galois representations, proposing a conjecture and providing evidence for their dependence.

Contribution

It conjectures that deformation spaces depend only on the Kisin variety and proves related properties linking Kisin varieties to Serre weights.

Findings

Kisin variety determines the cardinality of common Serre weights

Dependence on Kisin variety is nondecreasing

Number of weights is compatible with product decompositions

Abstract

Let $F$ be a finite unramified extension of $\mathbb Q\_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).