Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pullback of arithmetic theta series and its modularity for unitary Shimura curves

Qiao He, Yousheng Shi, Tonghai Yang

TL;DR

This work completes the modularity program for arithmetic theta functions on unitary Shimura curves in the delicate $U(1,1)$ case by embedding into higher-rank unitary varieties. It develops a precise pullback formula for arithmetic special divisors via a canonical morphism $\varphi_{\Lambda}$, relates two line bundles governing weight-1 modular forms by $\Omega=\omega\otimes \mathcal{O}(\mathrm{Exc})^{-1}$, and extends the deformation theory of special divisors to the boundary. Using the embedding trick and Kudla–Rapoport theory, it proves the modularity of the Kudla arithmetic theta function $\widehat{\Theta}_K(\tau)$ for $n=2$, which in turn implies the modularity of the Bruinier-type arithmetic theta function and yields a complete modularity statement for the arithmetic divisors in this setting. The paper also clarifies the interplay between the two Green-function normalizations (Bruinier and Kudla–Ehlen–Sankaran) and connects them through a detailed boundary analysis, enabling a robust application of Li’s embedding technique. Overall, it closes the gap in the Kudla program for unitary Shimura curves and provides a toolkit for boundary-deformation considerations in arithmetic intersection theory.

Abstract

This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modulartiy result on $U(1, 1)$ Shimura curves.

Pullback of arithmetic theta series and its modularity for unitary Shimura curves

TL;DR

This work completes the modularity program for arithmetic theta functions on unitary Shimura curves in the delicate case by embedding into higher-rank unitary varieties. It develops a precise pullback formula for arithmetic special divisors via a canonical morphism , relates two line bundles governing weight-1 modular forms by , and extends the deformation theory of special divisors to the boundary. Using the embedding trick and Kudla–Rapoport theory, it proves the modularity of the Kudla arithmetic theta function for , which in turn implies the modularity of the Bruinier-type arithmetic theta function and yields a complete modularity statement for the arithmetic divisors in this setting. The paper also clarifies the interplay between the two Green-function normalizations (Bruinier and Kudla–Ehlen–Sankaran) and connects them through a detailed boundary analysis, enabling a robust application of Li’s embedding technique. Overall, it closes the gap in the Kudla program for unitary Shimura curves and provides a toolkit for boundary-deformation considerations in arithmetic intersection theory.

Abstract

This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case not considered there. The main idea to embed a Shimura curve to many Shimura varieties for big , and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modulartiy result on Shimura curves.
Paper Structure (31 sections, 41 theorems, 257 equations)

This paper contains 31 sections, 41 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Unitary Shimura varieties and special divisors
  4. Unitary Shimura varieties
  5. Integral model
  6. Special divisors
  7. Cusp labels
  8. Degenerating abelian schemes
  9. Formal boundary charts
  10. Toroidal compactification
  11. Comparison of two line bundles
  12. Extensions of the line bundles to the boundary
  13. Constant term of the generating series
  14. Deformation of special divisors near the boundary
  15. Special divisors on formal boundary charts
  16. ...and 16 more sections

Key Result

Theorem 1.1

When $n =2$, $\widehat{\Theta}_B(\tau)$ and $\widehat{\Theta}_K(\tau)$ are modular forms for $\Gamma_0(D)$ of weight $2$, trivial character, and with values in $\widehat{\operatorname{CH}}_\mathbb{C}^1(\mathcal{S}^*)$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 76 more