Fred Diamond: Effective and minimal cones of weights for Hilbert modular forms

Abstract:

I’ll discuss some generalizations of the well-known fact that there are non non-zero modular forms of negative weight, even when working in characteristic p. In particular, for Hilbert modular forms associated to a totally real field of degree d, the weight is a d-tuple, all components of which are non-negative, if working in characteristic zero. But there are mod p Hilbert modular forms, called partial Hasse invariants, whose weight in some component is negative.

I’ll explain joint work with Kassaei (from 2017/2020) that shows the possible weights of non-zero Hilbert modular forms in characteristic p lie in the cone generated by the weights of these partial Hasse invariants (also proved independently by Goldring and Koskivirta). In fact we prove a stronger result (motivated by the relation with Galois representations) which asserts that any form whose weight lies outside a certain minimal cone is divisible by a partial Hasse invariant. I’ll also discuss a recent generalization of these results to forms on Goren–Oort strata of Hilbert modular varieties.