Monodromy and Irreducibility of Igusa Varieties

TL;DR

This paper investigates the structure of Igusa varieties and related strata in Shimura varieties of Hodge type, revealing new insights into their irreducible components and disproving a strong form of the Hecke orbit conjecture.

Contribution

It determines the irreducible components of Igusa varieties and central leaves, and extends methods to analyze Newton strata, combining recent advances in monodromy and Honda--Tate theory.

Findings

Disproves a strong version of the discrete Hecke orbit conjecture.

Identifies irreducible components of Igusa varieties and Newton strata.

Develops a unified approach using monodromy groups and Honda--Tate theory.

Abstract

We determine the irreducible components of Igusa varieties for Shimura varieties of Hodge type under a mild condition and use that to compute the irreducible components of central leaves. In particular, we show that a strong version of the discrete Hecke orbit conjecture is false in general. Our method combines recent work of D'Addezio on monodromy groups of compatible local systems with a generalisation of a method of Hida, using the Honda--Tate theory for Shimura varieties of Hodge type developed by Kisin--Madapusi--Shin. We also determine the irreducible components of Newton strata in Shimura varieties of Hodge type by combining our methods with recent work of Zhou--Zhu.