Serre weights and the Breuil-Mézard conjecture for modular forms

2020/04/16 by Wiersema, Hanneke
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2004.07587

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. 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).

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