Serre weights and the Breuil–Mézard conjecture for modular forms

Hanneke Wiersema

doi:10.5802/jtnb.1154

Hanneke Wiersema ¹

¹ King’s College London, Strand London, WC2R 2LS, United Kingdom

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 107-124.

Abstract
Résumé

Serre’s strong conjecture, now a theorem of Khare and Wintenberger, states that every two-dimensional continuous, odd, irreducible mod $p$ Galois representation $ρ$ arises from a modular form of a specific minimal weight $k (ρ)$ , level $N (ρ)$ and character $ϵ (ρ)$ . In this short paper we show that the minimal weight $k (ρ)$ is equal to a notion of minimal weight inspired by the recipe for weights introduced by Buzzard, Diamond and Jarvis in [4]. Moreover, using the Breuil–Mézard conjecture we show that both weight recipes are equal to the smallest $k \geq 2$ such that $ρ$ has a crystalline lift of Hodge–Tate type $(0, k - 1)$ .

La forme forte de la conjecture de Serre, démontrée par Khare et Wintenberger, assure que toute représentation galoisienne $ρ$ modulo $p$ , de dimension $2$ , continue, irréductible et impaire provient d’une forme modulaire de poids minimal $k (ρ)$ , de niveau $N (ρ)$ et de caractère $ϵ (ρ)$ prescrits. Dans ce court article, nous démontrons que le poids minimal $k (ρ)$ coïncide avec une autre notion de poids minimal, qui est inspirée par la recette pour les poids de $ρ$ introduite par Buzzard, Diamond et Jarvis dans [4]. De plus, en utilisant la conjecture de Breuil–Mézard, nous démontrons que le poids défini par ces recettes équivalentes est égal au plus petit entier $k \geq 2$ tel que $ρ$ possède un relèvement cristallin de type de Hodge–Tate $(0, k - 1)$ .

Received: 2020-04-03
Accepted: 2020-12-01
Published online: 2021-05-21

DOI: 10.5802/jtnb.1154

Classification: 11F80, 20C20
Keywords: Galois representations, Serre’s modularity conjecture, Breuil–Mézard conjecture

Author's affiliations:

Hanneke Wiersema ¹

¹ King’s College London, Strand London, WC2R 2LS, United Kingdom

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Hanneke Wiersema},
     title = {Serre weights and the {Breuil{\textendash}M\'ezard} conjecture for modular forms},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {107--124},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Hanneke Wiersema. Serre weights and the Breuil–Mézard conjecture for modular forms. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 107-124. doi : 10.5802/jtnb.1154. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/

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