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Combinatorics of Serre weights in the potentially Barsotti–Tate setting

Xavier Caruso, Agnès David and Ariane Mézard

Vol. 12 (2023), No. 1, 1–56

Let F be a finite unramified extension of p of degree f and ρ¯ be an absolutely irreducible mod p 2-dimensional representation of the absolute Galois group of F. Let t be a tame inertial type of level f of F. We conjecture that the deformation space parametrizing the potentially Barsotti–Tate liftings of ρ¯ having type t depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into (1)f and its shape stratification. We give evidence towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights 𝒟(t ,ρ¯) = 𝒟(t ) 𝒟(ρ¯). Additionally, we prove that this dependence is nondecreasing (the smaller the Kisin variety, the smaller the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

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$p$-adic Galois representations, deformations, Breuil–Mézard conjecture
Mathematical Subject Classification
Primary: 11S20
Received: 5 November 2021
Revised: 4 January 2023
Accepted: 18 January 2023
Published: 29 March 2023
Xavier Caruso
Université de Bordeaux
Agnès David
Université de Franche-Comté
Université de Rennes I
Campus de Beaulieu
Ariane Mézard
Département de Mathématiques et Applications
Ecole Normale Supérieure de Paris