On the geometry of affine Deligne-Lusztig varieties for quasi-split groups with very special leve

TL;DR

This paper studies the geometric structure of affine Deligne-Lusztig varieties with special level structures, establishing their dimensions and components, and applies these results to prove the Grothendieck conjecture for certain Shimura varieties and function fields.

Contribution

It provides a detailed geometric analysis of affine Deligne-Lusztig varieties with very special level, and proves the Grothendieck conjecture in this context.

Findings

Determined the dimension of affine Deligne-Lusztig varieties.

Classified their connected and irreducible components.

Proved the Grothendieck conjecture for specific Shimura varieties.

Abstract

In this paper we discuss the geometry of affine Deligne Lusztig varieties with very special level structure, determining their dimension and connected and irreducible components. As application, we prove the Grothendieck conjecture for Shimura varieties with very special level at $p$ and a function field analogue.