Divisibility of mod $p$ automorphic forms and the cone conjecture for certain Shimura varieties of Hodge-type

TL;DR

This paper demonstrates that for certain Hodge-type Shimura varieties, the cone of automorphic form weights is described by the stack of G-zips, and establishes divisibility properties of mod p automorphic forms by partial Hasse invariants.

Contribution

It proves the cone of weights is encoded by G-zip stacks and generalizes divisibility results of automorphic forms by partial Hasse invariants to new Shimura varieties.

Findings

The cone of weights is described by G-zip stacks.

Automorphic forms in certain weight regions are divisible by partial Hasse invariants.

Generalizes previous results on Hilbert modular forms.

Abstract

For several Hodge-type Shimura varieties of good reduction in characteristic $p$, we show that the cone of weights of automorphic forms is encoded by the stack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instances of a general conjecture formulated in previous papers by the authors. Furthermore, we prove in these cases that any mod $p$ automorphic form whose weight lies in a specific region of the weight space is divisible by a partial Hasse invariant. This generalizes to other Shimura varieties previous results of Diamond--Kassaei on Hilbert modular forms.