Igusa Stack for some exceptional Shimura Varieties

TL;DR

This paper proves a fiber product formula for certain Shimura varieties using Igusa stacks, enabling new local-global compatibility results and a vanishing theorem for their cohomology.

Contribution

It reformulates the construction of meta-unitary Shimura varieties via moduli stacks, extending previous work and applying the unipotent categorical local Langlands correspondence.

Findings

Validated the fiber product formula for non-abelian Shimura varieties.

Derived local-global compatibility results for these varieties.

Proved a vanishing theorem for the generic cohomology of meta-unitary Shimura varieties.

Abstract

We study the integral models of meta-unitary Shimura varieties through the lens of Scholze's fiber product conjecture. Reformulating Bultel's original construction in terms of moduli stacks of Shtukas and Igusa stacks, we prove the validity of the fiber product formula for this class of non-abelian type Shimura varieties, thereby generalizing the works of Zhang and Daniels, Van Hoften, Kim, and Zhang. We utilize this geometric description to derive local-global compatibility results and, adapting the strategy of Zhu and Yang, apply the unipotent categorical local Langlands correspondence to prove a general vanishing theorem for the generic part of the cohomology of meta-unitary Shimura varieties.