Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Connected components of special cycles on Shimura varieties

Keerthi Madapusi

TL;DR

The paper develops a p-adic, Chai–Hida–style framework to control irreducible components of special cycles on GSpin Shimura varieties by analyzing ordinary loci via $p$-Hecke correspondences and the ordinary Igusa tower. It shows that, under suitable rank conditions on the lattices, each irreducible component of a special fiber is the unique specialization of a component in the generic fiber, enabling irreducibility results for moduli of polarized K3 surfaces over all primes. The main mechanism combines deformation theory on the ordinary locus, the associated Igusa towers, and detailed group-scheme/quadric Grassmannian analyses to transfer connected-component data between characteristic $0$ and characteristic $p$. These component-control results feed into broader applications, including the modularity of generating series of higher codimension cycles on GSpin varieties. The work thus provides a robust bridge between integral geometry of special cycles and the global geometry of moduli spaces, with concrete consequences for K3 surface moduli and related arithmetic questions.

Abstract

I use methods of Chai-Hida and ordinary $p$-Hecke correspondences to study the set of irreducible components of special fibers of special cycles of sufficiently low codimension in integral models of GSpin Shimura varieties, and apply this to prove irreducibility results for the special fibers of the moduli of polarized K3 surfaces. These results are also applied in joint work with Howard on the modularity of generating series of higher codimension cycles on GSpin Shimura varieties.

Connected components of special cycles on Shimura varieties

TL;DR

The paper develops a p-adic, Chai–Hida–style framework to control irreducible components of special cycles on GSpin Shimura varieties by analyzing ordinary loci via -Hecke correspondences and the ordinary Igusa tower. It shows that, under suitable rank conditions on the lattices, each irreducible component of a special fiber is the unique specialization of a component in the generic fiber, enabling irreducibility results for moduli of polarized K3 surfaces over all primes. The main mechanism combines deformation theory on the ordinary locus, the associated Igusa towers, and detailed group-scheme/quadric Grassmannian analyses to transfer connected-component data between characteristic and characteristic . These component-control results feed into broader applications, including the modularity of generating series of higher codimension cycles on GSpin varieties. The work thus provides a robust bridge between integral geometry of special cycles and the global geometry of moduli spaces, with concrete consequences for K3 surface moduli and related arithmetic questions.

Abstract

I use methods of Chai-Hida and ordinary -Hecke correspondences to study the set of irreducible components of special fibers of special cycles of sufficiently low codimension in integral models of GSpin Shimura varieties, and apply this to prove irreducibility results for the special fibers of the moduli of polarized K3 surfaces. These results are also applied in joint work with Howard on the modularity of generating series of higher codimension cycles on GSpin Shimura varieties.
Paper Structure (25 sections, 42 theorems, 197 equations)

This paper contains 25 sections, 42 theorems, 197 equations.

Table of Contents

  1. Ordinary loci of Shimura varieties
  2. Background on Shimura varieties of Hodge type
  3. Deformation theory over the ordinary locus
  4. The ordinary Igusa tower
  5. The generic fiber of the Igusa tower
  6. Hecke correspondences
  7. Isogenies and $p$-Hecke correspondences in the generic fiber
  8. Ordinary isogenies and $p$-Hecke correspondences
  9. Comparison between the two notions of $p$-Hecke correspondences
  10. An argument of Chai and Hida
  11. The abstract setup
  12. Hecke action on connected components of Shimura varieties
  13. Kisin's analogue of Tate's theorem
  14. Monodromy over the ordinary loci of Shimura varieties
  15. Group schemes associated with quadratic lattices
  16. ...and 10 more sections

Key Result

Theorem A

For every prime $p$, the fiber $\mathsf{M}^{\circ}_{2d,\mathbb{F}_p}$ is geometrically irreducible.

Theorems & Definitions (159)

  • Theorem A
  • Theorem B
  • Lemma 1.1.4
  • proof
  • Remark 1.1.5
  • Lemma 1.1.6
  • proof
  • Remark 1.1.7
  • Remark 1.1.8
  • Definition 1.1.9
  • ...and 149 more