Minimal weights of mod p Galois representations

The strong form of Serre's conjecture states that every two-dimensional continuous, odd, irreducible mod p representation of the absolute Galois group of Q arises from a modular form of a specific minimal weight, level and character. In this talk we show the minimal weight is equal to a notion of minimal weight inspired by work of Buzzard, Diamond and Jarvis. Moreover, using the Breuil-Mézard conjecture we give a third interpretation of this minimal weight as the smallest k>1 such that the representation has a crystalline lift of Hodge-Tate type (0, k-1). After discussing the interplay between these three characterisations of minimal weight in the more general setting of Galois representations over totally real fields, we investigate its consequences for generalised Serre conjectures.

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https://us02web.zoom.us/j/89571187014