Serre weights and the Breuil-M\'{e}zard 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 $\rho$ arises from a modular form of a specific minimal weight $k(\rho)$, level $N(\rho)$ and character $\epsilon(\rho)$. In this short paper we show that the minimal weight $k(\rho)$ is equal to a notion of minimal weight inspired by the recipe for weights introduced by Buzzard, Diamond and Jarvis. Moreover, using the Breuil-M\'{e}zard conjecture we show that both weight recipes are equal to the smallest $k \geq 2$ such that $\rho$ has a crystalline lift of Hodge-Tate type $(0,k-1)$.