黑料社

Title:聽聽Cones of weights and Minimal cones of weights over Goren-Oort strata

Abstract:聽Let p be a prime. In previous joint work with F. Diamond, we introduced the notion of the minimal cone of weights: every Hecke system of eigenvalues on the space of mod p Hilbert modular forms is that of a mod p Hilbert eigenform with weight inside the minimal cone. As an immediate corollary of this work, we proved that the cone of weights of nonzero mod p Hilbert modular forms is generated by the weights of the partial Hasse invariants (i.e., the cone of weights equals the Hasse cone). For mod p automorphic forms over a general Shimura variety, it is yet unclear how to define the minimal cone of weights. In this talk, we will discuss a strategy for defining and calculating minimal cones of weights by considering the case of automorphic forms on the Goren-Oort strata of a Hilbert modular variety. We will define and determine the minimal cones of weights for all strata by determining their cones of weights (i.e., the cone generated by the weights of all nonzero automorphic forms on a stratum). In particular, we show that the cones of weights of strata are not generated by the weights of their respective partial Hasse invariants in general. Our proof hints at the idea that for a general Shimura variety X the discrepancy between the cone of weights and the Hasse cone could be explained by weights of the nonzero pullbacks of partial Hasse invariants on other Shimura varieties to which X maps. This is joint work with Fred Diamond.