Minimal weights of mod p Galois representations

Abstract: The strong form of Serre's conjecture states that every two-dimensional continuous, odd, irreducible mod p representation of the absolute Galois group of $\mathbb{Q}$ arises from a modular form of a specific minimal weight, level and character. In this talk we use modular representation theory to prove 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 $(0, k-1)$. Finally, we will report on work in progress where we study similar questions in the more general setting of mod $p$ Galois representations over a totally real field.

number theory

Audience: researchers in the topic

Cambridge Number Theory Seminar

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