Title: Geometric Serre weight conjectures, Hasse invariants and \Theta-operators

Abstract: Serre’s conjecture, now a theorem of Khare and Wintenberger, states that every odd, irreducible representation Gal(K/Q) --> GL(2,F_p) arises from a modular form. Furthermore it prescribes the minimal weight (at least 2) and level (prime to p) of such a form. I’ll recall this, along with a more geometric variant due to Edixhoven involving modular forms of weight one, and the relation with certain “weight-shifting” operators in characteristic p. Then I’ll discuss a generalization (joint with Sasaki) of the geometric variant and related weight-shifting phenomena in the context of Hilbert modular forms.