Monodromy and Irreducibility of Igusa Varieties
Pol van Hoften, Luciena Xiao Xiao
TL;DR
This work determines irreducible components of Igusa varieties over Shimura varieties of Hodge type under mild hypotheses, and uses this to describe the irreducible components of central leaves and Newton strata. The authors combine D'Addezio's results on monodromy with Honda–Tate–Kisin–Madapusi–Shin theory and Mantovan’s product formula to analyze both $\ell$-adic and $p$-adic monodromy, showing that the fibers of Igusa varieties over central leaves are controlled by $G^{\text{ab}}(\mathbb{Z}_p)$ and that certain monodromy groups equal kernels like $J_b'$. They establish surjectivity of certain maps on connected components, give a representation-theoretic count for Newton strata components via $\text{Dim}\,V_{\mu}(\lambda_b)_{\text{rel}}$, and prove that the strong discrete Hecke orbit conjecture fails in general while providing a full description of Igusa component structure in broad generality. The results advance understanding of $p$-adic automorphic forms on Hodge-type Shimura varieties and clarify the interaction between monodromy, Hecke actions, and the geometry of Newton and central leaves.
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.