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Kudla-Rapoport conjecture for unramified maximal parahoric level

Yu Luo

TL;DR

The paper proves the Kudla–Rapoport conjecture for unramified unitary groups with maximal parahoric level by embedding local data into a global arithmetic Siegel–Weil framework and performing a dimension-inductive argument based on modularity of generating series. It introduces weighted local density functions $oldsymbol{ alpha}_T(r,oldsymbol{ phi})$ and a correction mechanism with coefficients $eta_{oldsymbol{w},t}^{[h]}$ to match arithmetic intersections with derivatives of weighted densities, addressing non-invariance in the maximal parahoric setting. The proof proceeds via global integral models, derived cycles, and Eisenstein series, employing both non-archimedean and archimedean Siegel–Weil formulas and the geometric/analytic cancellation laws to achieve term-by-term modularity. The global construction of semi-global models, Green currents, and Eisenstein data provides a comprehensive framework linking local intersections to automorphic forms, and opens avenues for deeper levels and ramified cases. Overall, the result extends the arithmetic KR program to maximal parahoric level in the unitary setting and strengthens the bridge between arithmetic intersections and automorphic generating series.

Abstract

We prove the Kudla-Rapoport conjecture for unramified unitary groups with maximal parahoric level structure. Our approach differs from the local proof given in Li-W.Zhang. We reduce the conjecture to a global intersection problem using local-global compatibility. Then we apply an inductive procedure based on the modularity of generating series of global special divisors. This strategy follows the framework developed in the proof of the arithmetic fundamental lemma from W.Zhang and Mihatsch-W.Zhang and arithmetic transfer identities from Z.Zhang and Luo-Mihatsch-Z.Zhang.

Kudla-Rapoport conjecture for unramified maximal parahoric level

TL;DR

The paper proves the Kudla–Rapoport conjecture for unramified unitary groups with maximal parahoric level by embedding local data into a global arithmetic Siegel–Weil framework and performing a dimension-inductive argument based on modularity of generating series. It introduces weighted local density functions and a correction mechanism with coefficients to match arithmetic intersections with derivatives of weighted densities, addressing non-invariance in the maximal parahoric setting. The proof proceeds via global integral models, derived cycles, and Eisenstein series, employing both non-archimedean and archimedean Siegel–Weil formulas and the geometric/analytic cancellation laws to achieve term-by-term modularity. The global construction of semi-global models, Green currents, and Eisenstein data provides a comprehensive framework linking local intersections to automorphic forms, and opens avenues for deeper levels and ramified cases. Overall, the result extends the arithmetic KR program to maximal parahoric level in the unitary setting and strengthens the bridge between arithmetic intersections and automorphic generating series.

Abstract

We prove the Kudla-Rapoport conjecture for unramified unitary groups with maximal parahoric level structure. Our approach differs from the local proof given in Li-W.Zhang. We reduce the conjecture to a global intersection problem using local-global compatibility. Then we apply an inductive procedure based on the modularity of generating series of global special divisors. This strategy follows the framework developed in the proof of the arithmetic fundamental lemma from W.Zhang and Mihatsch-W.Zhang and arithmetic transfer identities from Z.Zhang and Luo-Mihatsch-Z.Zhang.

Paper Structure

This paper contains 61 sections, 50 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Kudla–Rapoport conjecture
  4. The arithmetic Siegel–Weil formula
  5. Proof strategy
  6. Open problems
  7. The structure of the paper
  8. Acknowledgments
  9. Notations and Conventions.
  10. Hermitian space
  11. Local data
  12. Global data
  13. Local Theory
  14. Rapoport-Zink spaces and special cycles
  15. Strict $O_{F_0}$-modules
  16. ...and 46 more sections

Key Result

Theorem 1.2.2

For any $0\leq h\leq n$ and for any weight vector $\mathbf{w}\xspace\in\{\mathcal{Z}\xspace,\mathcal{Y}\xspace\}^n$, let $\beta^{[h]}_{\mathbf{w}\xspace,t}$ be the explicit constants for $0 \leq t < h$, $t \equiv h-1 \pmod{2}$, given by Definition def:error-coeff. Then for any collection of linearly where $T=(\mathbf{u}\xspace,\mathbf{u}\xspace)$ and $\mathop{\mathrm{vol}}\nolimits(K^{[h]},dg)$ is

Theorems & Definitions (104)

  • Definition 1.2.1
  • Theorem 1.2.2: Theorem \ref{['thm:KR-conj']}, local Kudla-Rapoport conjecture
  • Remark 1.2.3
  • Example 1.2.4
  • Theorem 1.3.1: Arithmetic Siegel-Weil formula
  • Remark 1.4.1
  • Definition 2.2.1
  • Definition 2.3.1
  • Proposition 2.3.2
  • proof
  • ...and 94 more