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Spherical representations of unitary groups at ramified places and the arithmetic inner product formula

Zhuoni Chi

TL;DR

This work extends the arithmetic inner product formula to ramified places where the local root number is $-1$, by developing a unified treatment of spherical and almost-spherical representations for unitary groups at ramified places and constructing Krämer-type semi-global integral models. It combines a detailed local representation theory (including new bases, intertwining operators, and Satake-type descriptions for unimodular and (almost) modular lattices) with semi-global geometric models of unitary Shimura varieties and vanishing results for their cohomology, enabling precise matching of local zeta integrals with derivatives of doubling L-functions. The main payoff is a ramified generalization of the Kudla–Rapoport program: under parity and modularity hypotheses, the Beilinson–Bloch height pairing of arithmetic theta liftings is governed by $L'(\tfrac12,\pi)$ and explicitly computable local factors, even when the local root number equals $-1$. This provides new evidence for the arithmetic theta lifting philosophy in higher rank unitary groups and broadens the reach of the arithmetic inner product formula to ramified, non-split contexts with significant implications for the Beilinson–Bloch conjectures and special cycles on unitary Shimura varieties.

Abstract

In this article, we study admissible representations of even unitary groups over local fields, where the quadratic extension is ramified, with invariant vectors under the action of the stabilizer of a unimodular lattice and some properties of the corresponding integral model of unitary Shimura varieties. As a direct application, we are able to improve the arithmetic inner product formula so that the places with local root number \((-1)\) are allowed to be ramified.

Spherical representations of unitary groups at ramified places and the arithmetic inner product formula

TL;DR

This work extends the arithmetic inner product formula to ramified places where the local root number is , by developing a unified treatment of spherical and almost-spherical representations for unitary groups at ramified places and constructing Krämer-type semi-global integral models. It combines a detailed local representation theory (including new bases, intertwining operators, and Satake-type descriptions for unimodular and (almost) modular lattices) with semi-global geometric models of unitary Shimura varieties and vanishing results for their cohomology, enabling precise matching of local zeta integrals with derivatives of doubling L-functions. The main payoff is a ramified generalization of the Kudla–Rapoport program: under parity and modularity hypotheses, the Beilinson–Bloch height pairing of arithmetic theta liftings is governed by and explicitly computable local factors, even when the local root number equals . This provides new evidence for the arithmetic theta lifting philosophy in higher rank unitary groups and broadens the reach of the arithmetic inner product formula to ramified, non-split contexts with significant implications for the Beilinson–Bloch conjectures and special cycles on unitary Shimura varieties.

Abstract

In this article, we study admissible representations of even unitary groups over local fields, where the quadratic extension is ramified, with invariant vectors under the action of the stabilizer of a unimodular lattice and some properties of the corresponding integral model of unitary Shimura varieties. As a direct application, we are able to improve the arithmetic inner product formula so that the places with local root number \((-1)\) are allowed to be ramified.
Paper Structure (37 sections, 64 theorems, 189 equations)

This paper contains 37 sections, 64 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of this paper
  4. Acknowledgements
  5. Notations
  6. Notations for Local Theory
  7. Notations for Global Theory
  8. Spherical Representations and Almost Spherical Representations of Quasi-split Unitary groups
  9. Classical Results of Unramified Principal Series Representations
  10. Spherical Representations with Respect to the Standard Lattice
  11. Almost Spherical Representations with Respect to the Standard Lattice
  12. Spherical Representations with Respect to the Semi-standard Lattice
  13. Irreducibility of Unitary Principal Series Representations
  14. A Satake isomorphism
  15. Spherical Representations of Non-quasi-split Unitary groups
  16. ...and 22 more sections

Key Result

Theorem 1.5

Let $(\pi,\mathcal{V}_\pi)$ be as in Assumption asp:main with $r[\mathbb{F}:\mathbb{Q}]+|\mathtt{S}_\pi\cup\mathtt{S}'_\pi|$ oddWe do not need to assume li2021chow*Hypothesis 6.6 by kisin2021stable.. If $L'(\frac{1}{2},\pi)\neq0$, that is, $\mathrm{ord}_{s=\frac{1}{2}}L(s,\pi)=1$, then as long as $\ holds, where the colimit is taken over open compact subgroups $L_{\mathtt{R}}\subseteq H(\mathbb{F}

Theorems & Definitions (182)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Iwasawa decomposition, see, for example, cartier1979representations
  • ...and 172 more