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Kudla's modularity conjecture on integral models of orthogonal Shimura varieties

Benjamin Howard, Keerthi Madapusi

TL;DR

This work extends Kudla's modularity conjectures for special cycles to integral models of orthogonal Shimura varieties by constructing corrected cycle classes that better capture vertical components. The authors prove that the generating series of these corrected cycles is a holomorphic Siegel modular form with representation $\omega_{L,d}^*$, and they establish crucial compatibility with intersections, pullbacks, and the generic fiber. The approach blends moduli-theoretic constructions (via Kuga–Satake abelian schemes), low-codimension geometric analysis, Jacobi form technology, and derived algebraic geometry to define and study $\mathcal{C}(T,\mu)$, including a robust intersection theory and linear invariance. The results generalize Bruinier–Raum's generic-fiber modularity to the integral model, providing a solid arithmetic-geometric realization of Kudla's program with potential arithmetic applications.

Abstract

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.

Kudla's modularity conjecture on integral models of orthogonal Shimura varieties

TL;DR

This work extends Kudla's modularity conjectures for special cycles to integral models of orthogonal Shimura varieties by constructing corrected cycle classes that better capture vertical components. The authors prove that the generating series of these corrected cycles is a holomorphic Siegel modular form with representation , and they establish crucial compatibility with intersections, pullbacks, and the generic fiber. The approach blends moduli-theoretic constructions (via Kuga–Satake abelian schemes), low-codimension geometric analysis, Jacobi form technology, and derived algebraic geometry to define and study , including a robust intersection theory and linear invariance. The results generalize Bruinier–Raum's generic-fiber modularity to the integral model, providing a solid arithmetic-geometric realization of Kudla's program with potential arithmetic applications.

Abstract

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.
Paper Structure (37 sections, 58 theorems, 521 equations)

This paper contains 37 sections, 58 theorems, 521 equations.

Table of Contents

  1. Introduction
  2. Modularity on the generic fiber
  3. Modularity on the integral model
  4. Outline of the paper
  5. Acknowledgements
  6. The Shimura variety and its special cycles
  7. The Shimura variety
  8. Integral models and special cycles
  9. Special cycles as Shimura varieties
  10. Basic properties of special cycles
  11. Special cycles of low codimension
  12. Connectedness in low codimension
  13. Geometric properties in low codimension: the self-dual case
  14. Geometric properties in low codimension: the general case
  15. Modularity in low codimension
  16. ...and 22 more sections

Key Result

Theorem A

The formal generating series converges to a Siegel modular form of weight $\frac{n}{2} +1$ and representation Convergence and modularity are understood in the following sense: for any $\mathbb Q$-linear functional $\iota : \mathrm{CH}^{d}(M) \to \mathbb C$, the formal generating series with coefficients in $S_{L,d}^*$ is the $q$-expansion of a holomorphic Siegel modular form of the stated weigh

Theorems & Definitions (149)

  • Theorem A: Bruinier-Raum
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Corollary G
  • Remark 2.1.1
  • Remark 2.1.2
  • Definition 2.2.1
  • Remark 2.2.2
  • ...and 139 more