Mathematics Seminar
Date: Tuesday, 21 January 2025
Time: 3:30 PM
Venue: Seminar Hall
Effective and minimal cones of weights for Hilbert modular forms (joint with P. Kassaei)
Fred Diamond
King's College, London.
21-01-25

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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 discuss joint work with Kassaei 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. 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 generalization of these results to forms on Goren-Oort strata of Hilbert modular varieties.