Michael Schein

Weights in Serre's conjecture for Hilbert modular forms

Let F be a totally real field, and consider a Galois representation ρ: Gal(Fbar/F) → GL2(Fpbar). If p is unramified in F, then Buzzard, Diamond, and Jarvis formulated an analogue of Serre's conjecture that specifies when ρ should be modular and predicts the modular weights. We will present an extension of their conjecture to arbitrary F in the case where ρ is tamely ramified at p and discuss a theorem stating, under certain conditions in this setting, that modular weights are contained in the list of predicted weights. We will also present a result of the same form, but proved by entirely different methods, for representations ρ: Gal(Qbar / Q) → GLn(Fpbar). It provides evidence for the conjecture stated in Florian Herzig's talk.