Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weighted special cycles on Rapoport--Zink spaces with almost self-dual level

Qiao He, Zhiyu Zhang, Baiqing Zhu

Abstract

We introduce a ``vector valued'' version of special cycles on GSpin Rapoport--Zink spaces with almost self-dual level in the context of the Kudla program, with certain linear invariance and local modularity features. They are local analogs of special cycles on GSpin Shimura varieties with almost self-dual parahoric level (e.g. Siegel threefolds with paramodular level). We establish local arithmetic Siegel--Weil formulas relating arithmetic intersection numbers of these special cycles and derivatives of certain local Whittaker functions in any dimension. The proof is based on a reduction formula for cyclic quadratic lattices.

Weighted special cycles on Rapoport--Zink spaces with almost self-dual level

Abstract

We introduce a ``vector valued'' version of special cycles on GSpin Rapoport--Zink spaces with almost self-dual level in the context of the Kudla program, with certain linear invariance and local modularity features. They are local analogs of special cycles on GSpin Shimura varieties with almost self-dual parahoric level (e.g. Siegel threefolds with paramodular level). We establish local arithmetic Siegel--Weil formulas relating arithmetic intersection numbers of these special cycles and derivatives of certain local Whittaker functions in any dimension. The proof is based on a reduction formula for cyclic quadratic lattices.

Paper Structure

This paper contains 47 sections, 50 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Strategy of the proof
  5. Structure of the paper
  6. Notations
  7. Acknowledgement
  8. Analytic side
  9. Reductions on quadratic lattices
  10. Notations on quadratic lattices
  11. Sublattices of self-dual quadratic lattices
  12. Cyclic quadratic lattices
  13. Vector valued Whittaker functions
  14. Local densities
  15. Counting lattices
  16. ...and 32 more sections

Key Result

Proposition 1.2.1

The special cycle $\mathcal{Y}^{\textup{bl}}(x)\subset\mathcal{M}_L$ is an effective Cartier divisor. Moreover, we have the following equality as Cartier divisors on $\mathcal{M}_L$: where $\textup{Exc}_L(x)=\bigsqcup\limits_{z\in\mathcal{N}_L^{\textup{nfs}}\cap\mathcal{Y}(x)}\pi_L^{-1}(z)$.

Theorems & Definitions (119)

  • Proposition 1.2.1: Proposition \ref{['decom-Y']}
  • Theorem 1.2.2: Theorem \ref{['main']}
  • Lemma 2.2.1
  • proof
  • Definition 2.3.1
  • Lemma 2.3.2
  • proof
  • Lemma 2.3.3
  • proof
  • Lemma 2.3.4
  • ...and 109 more