Serre weights and the Breuil–Mézard conjecture for modular forms
Journal de Théorie des Nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 107-124.

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 k2 tel que ρ possède un relèvement cristallin de type de Hodge–Tate (0,k-1).

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 k2 such that ρ has a crystalline lift of Hodge–Tate type (0,k-1).

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DOI : https://doi.org/10.5802/jtnb.1154
Classification : 11F80,  20C20
Mots clés : Galois representations, Serre’s modularity conjecture, Breuil–Mézard conjecture
@article{JTNB_2021__33_1_107_0,
     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},
     volume = {33},
     number = {1},
     year = {2021},
     doi = {10.5802/jtnb.1154},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/}
}
Hanneke Wiersema. Serre weights and the Breuil–Mézard conjecture for modular forms. Journal de Théorie des Nombres de Bordeaux, Tome 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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