Serre weights and the Breuil–Mézard conjecture for modular forms

Abstract

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≥2 such that ρ has a crystalline lift of Hodge–Tate type (0,k-1).