Skip to main content
SpringerLink
Log in

Families of canonical local periods on spherical varieties

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We consider the variation of canonical local periods on spherical varieties proposed by Sakellaridis–Venkatesh in families. We formulate conjectures for the rationality and meromorphic property of canonical local periods and establish these conjectures for strongly tempered spherical G-varieties without type N-roots when G is split.

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

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Beuzart-Plessis, R., Chaudouard, P., Zydor, M.: The global Gan–Gross–Prasad conjecture for unitary groups: the endoscopic case. Publ. Math. Inst. Hautes Études Sci. 135, 183–336 (2022)

    Article  MathSciNet  Google Scholar 

  2. Blanc, P., Delorme, P.: Vecteurs distributions \(H\)-invariants de représentations induites, pour un espace symétrique réductif \(p\)-adique \(G/H\). Ann. Inst. Fourier (Grenoble) 58(1), 213–261 (2008)

    Article  MathSciNet  Google Scholar 

  3. Bertolini, M., Darmon, H., Prasanna, K.: Generalized Heegner cycles and \(p\)-adic Rankin L-series, with an appendix by B. Conrad. Duke Math. J. 162, 1033–1148 (2013)

    Article  MathSciNet  Google Scholar 

  4. Beuzart-Plessis, R., Liu, Y., Zhang, W., Zhu, X.: Isolation of cuspidal spectrum, with application to the Gan–Gross–Prasad conjecture. Ann. Math. (2) 194(2), 519–584 (2021)

    Article  MathSciNet  Google Scholar 

  5. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. (French) Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)

  6. Cai, L., Fan, Y.: Spherical Characters in Families: the unitary Gan-Gross-Prasad case. arXiv:2305.11555

  7. Casselman, W.A.: The unramified principal series of \(p\)-adic groups. I. The spherical function. Compos. Math. 40(3), 387–406 (1980)

    MathSciNet  Google Scholar 

  8. Casselman, W.A.: Introduction to the theory of admissible representations of \(p\)-adic reductive groups. Draft (1995)

  9. Casselman, W.A.: Remarks on Macdonald’s book on \(p\)-adic spherical functions (2012)

  10. Cogdell, J., Piatetski-Shapiro, I.I.: Derivatives and L-functions for \({\rm GL }(n)\). (English summary) Representation theory, number theory, and invariant theory. Progr. Math. Birkhäuser/Springer, Cham, 323, 115–173 (2017)

  11. Dat, J.-F., Helm, D., Kurinczuk, R., Moss, G.: Finiteness for Hecke algebras of \(p\)-adic groups. arXiv:2203.04929

  12. Disegni, D.: Local Langlands correspondence, local factors, and zeta integrals in analytic families. J. Lond.Math. Soc. 101, 735–764 (2020)

    Article  MathSciNet  Google Scholar 

  13. Disegni, D.: The universal \(p\)-adic Gross–Zagier formula. Invent. Math. 230(2), 509–649 (2022)

    Article  MathSciNet  Google Scholar 

  14. Disegni, D.: \(p\)-adic \(L\)-functions via local-global interpolations: the case of \({\rm GL}_2\times {\rm GU}(1)\). Can. J. Math. 75(3), 965–1017 (2023)

  15. Emerton, M., Helm, D.: The local Langlands correspondence for \({\rm GL}_n\) in families. (English, French summary) Ann. Sci. Éc. Norm. Supér. (4) 47(4), 655–722 (2014)

  16. Feigon, B., Lapid, E., Offen, O.: On representations distinguished by unitary groups. (English summary) Publ. Math. Inst. Hautes Études Sci. 115, 185–323 (2012)

  17. Girsch, J.: The doubling method in algebraic families. arXiv:2102.09062

  18. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977). ISBN:0-387-90244-9

  19. Harris, M.: Square root \(p\)-adic \(L\)-functions, I: construction of a one-variable measure. Tunis. J. Math. 3(4), 657–688 (2021)

  20. Helm, D.: Whittaker models and the integral Bernstein center for \({\rm GL}_n\). (English summary) Duke Math. J. 165(9), 1597–1628 (2016)

  21. Hsieh, M.L.: Hida families and \(p\)-adic triple product L-functions. (English summary) Am. J. Math. 143(2), 411–532 (2021)

  22. Ichino, A.: Trilinear forms and the central values of triple product \(L\)-functions. (English summary) Duke Math. J. 145(2), 281–307 (2008)

  23. Knop, F., Schalke, B.: The dual group of a spherical variety. Trans. Mosc. Math. Soc. 78, 187–216 (2017)

    Article  MathSciNet  Google Scholar 

  24. Li, W.-W.: Zeta Integrals, Schwartz Spaces and Local Functional Equations. Springer, Berlin (2018)

    Book  Google Scholar 

  25. Liu, Y., Zhang, S., Zhang, W.: A \(p\)-adic Waldspurger formula. Duke. Math. J. 167, 743–833 (2018)

    Article  MathSciNet  Google Scholar 

  26. Moss, G.: Interpolating local constants in families. Math. Res. Lett. 23, 1789–1817 (2016)

    Article  MathSciNet  Google Scholar 

  27. Moss, G.: Gamma factors of pairs and a local converse theorem in families. Int. Math. Res. Not. 2016(16), 4903–4936 (2016)

    Article  MathSciNet  Google Scholar 

  28. Prasad, D.: Ext-analogues of branching laws. (English summary). In: Proceedings of the International Congress of Mathematicians Rio de Janeiro 2018, vol. II. Invited lectures, pp. 1367–1392. World Sci. Publ., Hackensack (2018)

  29. Prasad, D.: Generalizing the MVW involution, and the contragredient. Trans. Am. Math. Soc. 372(1), 615–633 (2019)

    Article  MathSciNet  Google Scholar 

  30. Sakellaridis, Y.: On the unramified spectrum of spherical varieties over \(p\)-adic fields. (English summary) Compos. Math. 144(4), 978–1016 (2008)

  31. Sakellaridis, Y., Venkatesh, A.: Periods and harmonic analysis on spherical varieties. Asterisque 396, 360 (2017)

  32. Silberger, A.J.: Introduction to Harmonic Analysis on Reductive \(p\)-Adic Groups. Based on Lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. Mathematical Notes, vol. 23. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1979)

  33. Authors: Stack projects. https://stacks.math.columbia.edu

  34. Wan, C., Zhang, L.: Periods of automorphic forms associated to strongly tempered spherical varieties. arXiv:2102.03695

  35. Zhang, W.: Automorphic period and the central value of Rankin–Selberg L-function. J. Am. Math. Soc. 27, 541–612 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We express our sincere gratitude to Prof. Y. Tian for his consistent encouragement, and to Prof. A. Burungale for carefully reading earlier versions of the article. We are grateful to the anonymous referee whose comments helped us improve the article. In particular, we thank the referee for pointing out a gap in an earlier version and Prof. G. Moss for generously suggesting Proposition 4.20 to fix it. We thank Prof. J.-F. Dat for kindly answering our question on Jacquet modules and thank Prof. J. Yang for discussions on global applications. L. Cai is partially supported by NSFC Grant no. 11971254.

Author information

Authors and Affiliations

  1. Academy for Multidisciplinary Studies, Beijing National Center for Applied Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China

    Li Cai & Yangyu Fan

Authors
  1. Li Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yangyu Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yangyu Fan.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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

Cai, L., Fan, Y. Families of canonical local periods on spherical varieties. Math. Ann. 389, 209–252 (2024). https://doi.org/10.1007/s00208-023-02642-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-023-02642-6

Search

Navigation