Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mixed mock modularity of special divisors

Philip Engel, François Greer, Salim Tayou

TL;DR

The paper proves that the generating series of special divisors on toroidal compactifications of orthogonal Shimura varieties forms a mixed mock modular form, with an explicit non-holomorphic completion built from theta-series attached to boundary rays. By combining Kudla–Millson modularity for the open part with precise boundary-intersection computations on toric boundary strata and a Vignéras-type modularity criterion for the resulting theta-series, the authors show the full series is a weakly holomorphic modular form of weight $1+ rac{n}{2}$ and representation $ar{ ho}_L$, with rational boundary coefficients. A key technical advance is a splitting of $H_2(X_ ext{Γ}^ ext{Σ}, ext{Q})$ into interior and boundary pieces, allowing modularity to be established boundary-by-boundary. The results extend the Kudla–Millson framework to compactified moduli spaces, clarify the role of boundary contributions via explicit ray theta-series, and offer a foundation for potential generalizations to higher-codimension cycles and other compactifications.

Abstract

We prove that the generating series of special divisors in toroidal compactifications of orthogonal Shimura varieties is a mixed mock modular form. More precisely, we find an explicit completion using theta series associated to rays in the cone decomposition. The proof relies on intersection theory at the boundary of the Shimura variety.

Mixed mock modularity of special divisors

TL;DR

The paper proves that the generating series of special divisors on toroidal compactifications of orthogonal Shimura varieties forms a mixed mock modular form, with an explicit non-holomorphic completion built from theta-series attached to boundary rays. By combining Kudla–Millson modularity for the open part with precise boundary-intersection computations on toric boundary strata and a Vignéras-type modularity criterion for the resulting theta-series, the authors show the full series is a weakly holomorphic modular form of weight and representation , with rational boundary coefficients. A key technical advance is a splitting of into interior and boundary pieces, allowing modularity to be established boundary-by-boundary. The results extend the Kudla–Millson framework to compactified moduli spaces, clarify the role of boundary contributions via explicit ray theta-series, and offer a foundation for potential generalizations to higher-codimension cycles and other compactifications.

Abstract

We prove that the generating series of special divisors in toroidal compactifications of orthogonal Shimura varieties is a mixed mock modular form. More precisely, we find an explicit completion using theta series associated to rays in the cone decomposition. The proof relies on intersection theory at the boundary of the Shimura variety.
Paper Structure (23 sections, 22 theorems, 152 equations)

This paper contains 23 sections, 22 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Comparison with earlier work
  4. Organization of the paper
  5. Acknowledgement
  6. Weil representation and vector-valued modular forms
  7. Weil representation
  8. Weak harmonic Maass forms and mixed mock modular forms
  9. Zagier's Eisenstein series and generalizations
  10. Vignéras' modularity criterion
  11. Hyperbolic lattices
  12. Case of non-isotropic vectors
  13. Case of isotropic vectors
  14. Orthogonal Shimura varieties
  15. Baily-Borel compactification
  16. ...and 8 more sections

Key Result

Theorem 1.1

The generating series of cohomology classes of Zariski closures $Z(\beta,m):= \overline{Z^o(\beta,m)}$, valued in $H^2(X_\Gamma^\Sigma,\mathbb{Q})\otimes \mathbb{C}[L^\vee/L]$, is a mixed mock modular form of weight $1+n/2$ with respect to $\overline{\rho}_L$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 39 more