Skip to main content
SpringerLink
Log in

A proof of the Kudla–Rapoport conjecture for Krämer models

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We prove the Kudla–Rapoport conjecture for Krämer models of unitary Rapoport–Zink spaces at ramified places. It is a precise identity between arithmetic intersection numbers of special cycles on Krämer models and modified derived local densities of hermitian forms. As an application, we relax the local assumptions at ramified places in the arithmetic Siegel–Weil formula for unitary Shimura varieties, which is in particular applicable to unitary Shimura varieties associated to unimodular hermitian lattices over imaginary quadratic fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We refrain from using the terminology self-dual in the ramified case to avoid possible confusion with a lattice \(L\) such that \(L=L^{\vee}\), where \(L^{\vee}\) is the dual lattice with respect to the underlying quadratic form, see §4.2.

  2. Notice that the vertex lattice \(\Lambda \) in the sense of [9, 26] or [10] corresponds to \(\Lambda ^{\sharp}\) in our convention.

References

  1. Ahsendorf, T., Cheng, C., Zink, T.: \(\mathcal{O}\)-Displays and \(\pi \)-divisible formal \(\mathcal{O}\)-modules. J. Algebra 457, 129–193 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruinier, J., Howard, B., Kudla, S.S., Rapoport, M., Yang, T.: Modularity of generating series of divisors on unitary Shimura varieties I. Astérisque 421, 7–125 (2020)

    Article  MathSciNet  Google Scholar 

  3. Cameron, P.: Lecture notes of advanced combinatorics mt5821. http://www-groups.mcs.st-andrews.ac.uk/~pjc/Teaching/MT5821/1/l6.pdf

  4. Cho, S.: Special cycles on unitary Shimura varieties with minuscule parahoric level structure. Math. Ann. 384, 1747–1813 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cho, S., Yamauchi, T.: A reformulation of the Siegel series and intersection numbers. Math. Ann. 377(3), 1757–1826 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Disegni, D., Liu, Y.: A \(p\)-adic arithmetic inner product formula. arXiv e-prints, (2022). arXiv:2204.09239

  7. Garcia, L.E., Sankaran, S.: Green forms and the arithmetic Siegel-Weil formula. Invent. Math. 215(3), 863–975 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gross, B.: On canonical and quasi-canonical liftings. Invent. Math. 84(2), 321–326 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. He, Q., Shi, Y., Yang, T.: The Kudla-Rapoport conjecture at a ramified prime for \(U(1,1)\). Trans. Am. Math. Soc. 376, 2257–2291 (2023)

    MathSciNet  MATH  Google Scholar 

  10. He, Q., Shi, Y., Yang, T.: Kudla-Rapoport conjecture for Krämer models. Compos. Math. 159, 1673–1740 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  11. Howard, B.: Linear invariance of intersections on unitary Rapoport–Zink spaces. Forum Math. 31, 1265–1281 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Krämer, N.: Local models for ramified unitary groups. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 73, pp. 67–80. Springer, Berlin (2003)

    Google Scholar 

  13. Kudla, S.: Central derivatives of Eisenstein series and height pairings. Ann. Math. 146(3), 545–646 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kudla, S.S.: Special cycles and derivatives of Eisenstein series. In: Heegner Points and Rankin \(L\)-Series. Math. Sci. Res. Inst. Publ., vol. 49, pp. 243–270. Cambridge University Press, Cambridge (2004)

    Chapter  Google Scholar 

  15. Kudla, S.S., Rapoport, M.: Height pairings on Shimura curves and \(p\)-adic uniformization. Invent. Math. 142(1), 153–223 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kudla, S.S., Rapoport, M.: Special cycles on unitary Shimura varieties I. Unramified local theory. Invent. Math. 184(3), 629–682 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kudla, S.S., Rapoport, M.: Special cycles on unitary Shimura varieties II: global theory. J. Reine Angew. Math. 697, 91–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C., Liu, Y.: Chow groups and \(L\)-derivatives of automorphic motives for unitary groups. Ann. Math. (2) 194(3), 817–901 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, C., Liu, Y.: Chow groups and \(L\)-derivatives of automorphic motives for unitary groups, II. Forum Math. Pi 10, e5 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, C., Zhang, W.: Kudla–Rapoport cycles and derivatives of local densities. J. Am. Math. Soc. 35(3), 705–797 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, C., Zhang, W.: On the arithmetic Siegel–Weil formula for GSpin Shimura varieties. Invent. Math. 228(3), 1353–1460 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, Y.: Arithmetic theta lifting and \(L\)-derivatives for unitary groups, I. Algebra Number Theory 5(7), 849–921 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mihatsch, A.: Relative unitary RZ-spaces and the arithmetic fundamental lemma. J. Inst. Math. Jussieu 21(1), 241–301 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pappas, G.: On the arithmetic moduli schemes of PEL Shimura varieties. J. Algebraic Geom. 9(3), 577 (2000)

    MathSciNet  MATH  Google Scholar 

  25. Rapoport, M., Terstiege, U., Zhang, W.: On the arithmetic fundamental lemma in the minuscule case. Compos. Math. 149(10), 1631–1666 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rapoport, M., Terstiege, U., Wilson, S.: The supersingular locus of the Shimura variety for \(\mathrm{GU}(1,n-1)\) over a ramified prime. Math. Z. 276(3–4), 1165–1188 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rapoport, M., Smithling, B., Zhang, W.: Arithmetic diagonal cycles on unitary Shimura varieties. Compos. Math. 156(9), 1745–1824 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rapoport, M., Smithling, B., Zhang, W.: On Shimura varieties for unitary groups. Pure Appl. Math. Q. 17(2), 773–837 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shi, Y.: Special cycles on unitary Shimura curves at ramified primes. Manuscr. Math., 1–70 (2022). https://doi.org/10.1007/s00229-022-01412-z

    Article  MATH  Google Scholar 

  30. Shi, Y.: Special cycles on the basic locus of unitary Shimura varieties at ramified primes. Algebra Number Theory (2018, in press)

  31. Siegel, C.L.: Über die analytische Theorie der quadratischen Formen. Ann. Math. (2) 36(3), 527–606 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  32. Siegel, C.L.: Indefinite quadratische Formen und Funktionentheorie. I. Math. Ann. 124, 17–54 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  33. Soulé, C., Abramovich, D., Burnol, J.F., Kramer, J.K.: Lectures on Arakelov Geometry. Cambridge Studies in Advanced Mathematics, vol. 33. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  34. Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety of GU(\(1, n-1\)) II. Invent. Math. 184(3), 591–627 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Weil, A.: Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math. 113, 1–87 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, W.: Weil representation and arithmetic fundamental lemma. Ann. Math. 193(3), 863–978 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the referee for the careful reading and many valuable suggestions. Revision of the work was done while Q.H., C.L. and T.Y. attended the “Algebraic Cycles, \(L\)-Values, and Euler Systems” program held in MSRI in Spring 2023. We would like to thank MSRI for the excellent work condition, financial support, and hospitality.

Funding

C. L. was partially supported by the NSF grant DMS-2101157. T.Y. was partially supported by the Dorothy Gollmar Chair’s Fund and Van Vleck research fund. Q.H. and T.Y. are partially supported by a graduate school grant of UW-Madison.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall, Madison, WI, 53706, USA

    Qiao He & Tonghai Yang

  2. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Chao Li

  3. School of Mathematical Sciences, Zhejiang University, 866 Yuhangtang Rd, Hangzhou, 310058, P.R. China

    Yousheng Shi

Authors
  1. Qiao He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yousheng Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tonghai Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tonghai Yang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, Q., Li, C., Shi, Y. et al. A proof of the Kudla–Rapoport conjecture for Krämer models. Invent. math. 234, 721–817 (2023). https://doi.org/10.1007/s00222-023-01209-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-023-01209-1

Search

Navigation