Hilbert modular forms and geometric modularity in quadratic case

Abstract: Let \rho: G_\Q \rightarrow \GL_2(\Fpbar) be a continuous, odd, irreducible representation. The weight part of Serre's conjecture predicts the minimal weight k (\geq 2) such that \rho arises from a modular eigenform of weight k. It is refined by Edixhoven to include the weight one forms by viewing mod p modular forms as sections of certain line bundles on the special fibre of a modular curve. One of the directions to generalise the weight part of Serre's conjecture is replacing Q with a totally real field F and replacing modular forms with Hilbert modular forms. A conjecture in this setting is formulated by Buzzard, Diamond and Jarvis, where we have the notion of algebraic modularity. On the other hand, a generalisation of Edixhoven's refinement is considered by Diamond and Sasaki, where we have the notion of geometric modularity. I will discuss the relation between algebraic and geometric modularity and show their consistency for the weights in a certain cone, under the assumption that F is a real quadratic field in which p is unramified.

number theory

Audience: researchers in the topic

London number theory seminar

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