Weights of mod $p$ automorphic forms and partial Hasse invariants

TL;DR

This paper introduces the zip cone of weights for mod p automorphic forms on G-zips, conjectures its equality with the automorphic form weight cone for certain Shimura varieties, and proves key inclusions and generation properties.

Contribution

It defines the zip cone for G-zips, proves the inclusion of characteristic 0 automorphic form weights, and characterizes when the cone is generated by partial Hasse invariants.

Findings

The zip cone is contained in the weight cone of characteristic 0 automorphic forms.

The paper characterizes when the zip cone is generated by partial Hasse invariants.

Provides evidence for the conjectured equality of cones in automorphic forms theory.

Abstract

For a connected, reductive group $G$ over a finite field endowed with a cocharacter $\mu$, we define the zip cone of $(G,\mu)$ as the cone of all possible weights of mod $p$ automorphic forms on the stack of $G$-zips. This cone is conjectured to coincide with the cone of weights of characteristic $p$ automorphic forms for Hodge-type Shimura varieties of good reduction. We prove in full generality that the cone of weights of characteristic $0$ automorphic forms is contained in the zip cone, which gives further evidence to this conjecture. Furthermore, we determine exactly when the zip cone is generated by the weights of partial Hasse invariants, which is a group-theoretical generalization of a result of Diamond--Kassaei and Goldring--Koskivirta.