Serre weights and the Breuil–Mézard conjecture for modular forms

Wiersema, Hanneke

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 $\rho$ modulo $p$, de dimension $2$, continue, irréductible et impaire provient d’une forme modulaire de poids minimal $k\left(\rho \right)$, de niveau $N\left(\rho \right)$ et de caractère $ϵ\left(\rho \right)$ prescrits. Dans ce court article, nous démontrons que le poids minimal $k\left(\rho \right)$ coïncide avec une autre notion de poids minimal, qui est inspirée par la recette pour les poids de $\rho$ 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 $k\ge 2$ tel que $\rho$ 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 $\rho$ arises from a modular form of a specific minimal weight $k\left(\rho \right)$, level $N\left(\rho \right)$ and character $ϵ\left(\rho \right)$. In this short paper we show that the minimal weight $k\left(\rho \right)$ 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 $k\ge 2$ such that $\rho$ has a crystalline lift of Hodge–Tate type $(0,k-1)$.

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

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