Advertisement

Inventiones mathematicae

, Volume 188, Issue 1, pp 197–252 | Cite as

On arithmetic fundamental lemmas

  • Wei Zhang
Article

Abstract

We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura varieties. As a crucial ingredient, we formulate a conjectural arithmetic fundamental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.

Keywords

Unitary Group Double Coset Algebraic Cycle Fundamental Lemma Orbital Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beilinson, A.: Height pairing between algebraic cycles. In: Current Trends in Arithmetical Algebraic Geometry, Arcata, CA, 1985. Contemp. Math., vol. 67, pp. 1–24. Am. Math. Soc., Providence (1987) CrossRefGoogle Scholar
  2. 2.
    Bloch, S.: Height pairings for algebraic cycles. J. Pure Appl. Algebra 34(2–3), 119–145 (1984). Proceedings of the Luminy Conference on Algebraic K-Theory (Luminy, 1983) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. Preprint http://www.math.harvard.edu/~gross/preprints/
  4. 4.
    Ginzburg, D., Jiang, D., Rallis, S.: On the nonvanishing of the central value of the Rankin–Selberg L-functions. J. Am. Math. Soc. 17(3), 679–722 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ginzburg, D., Jiang, D., Rallis, S.: On the nonvanishing of the central value of the Rankin–Selberg L-functions. II. In: Automorphic Representations, L-functions and Applications: Progress and Prospects, pp. 157–191 Google Scholar
  6. 6.
    Gordon, J.: Transfer to characteristic zero—appendix to “The fundamental lemma of Jacquet–Rallis in positive characteristics” by Zhiwei Yun. arXiv:1005.0610
  7. 7.
    Gross, B.H.: On canonical and quasicanonical liftings. Invent. Math. 84(2), 321–326 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gross, B.H., Prasad, D.: On the decomposition of a representation of SO n when restricted to SOn−1. Can. J. Math. 44(5), 974–1002 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gross, B.H., Prasad, D.: On irreducible representations of SO 2n+1×SO 2m. Can. J. Math. 46(5), 930–950 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gross, B., Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84(2), 225–320 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Harris, M., Labesse, J.-P.: Conditional base change for unitary groups. Asian J. Math. 8(4), 653–684 (2004) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ichino, A.: Trilinear forms and the central values of triple product L-functions. Duke Math. J. 145(2), 281–307 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jacquet, H., Rallis, S.: On the Gross–Prasad conjecture for unitary groups. http://www.math.columbia.edu/~hj/GrossPrasadFinal.pdf
  14. 14.
    Kottwitz, R.E.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 5(2), 373–444 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kudla, S.: Central derivatives of Eisenstein series and height pairings. Ann. Math. (2) 146(3), 545–646 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kudla, S., Rapoport, M.: Special cycles on unitary Shimura varieties I. Unramified local theory. Invent. Math. 184(3), 629–682 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kudla, S., Rapoport, M.: Special cycles on unitary Shimura varieties II: global theory. arXiv:0912.3758
  18. 18.
    Kudla, S.S., Rapoport, M., Yang, T.: Modular Forms and Special Cycles on Shimura Curves. Annals of Mathematics Studies, vol. 161. Princeton University Press, Princeton (2006). x+373 pp zbMATHGoogle Scholar
  19. 19.
    Y., Liu, Arithmetic theta lifting and central derivatives of L-functions for U(1,1). Preprint (2009) Google Scholar
  20. 20.
    Rallis, S., Schiffmann, G.: Multiplicity one conjectures. arXiv:0705.2168
  21. 21.
    Rapoport, M., Zink, Th.: Period Spaces for p-Divisible Groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996) Google Scholar
  22. 22.
    Vollaard, I.: Endomorphisms of quasi-canonical lifts. Astérisque 312, 105–112 (2007) MathSciNetGoogle Scholar
  23. 23.
    Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety for GU(1,n−1), II. Invent. Math. 3, 591–627 (2011) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Waldspurger, J.: Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compos. Math. 54(2), 173–242 (1985) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Yuan, X., Zhang, S.-W., Zhang, W.: Gross–Zagier formula on Shimura curves. Ann. Math. Stud. (submitted) Google Scholar
  26. 26.
    Yuan, X., Zhang, S.-W., Zhang, W.: Triple product L-series and Gross–Schoen cycles. Preprint Google Scholar
  27. 27.
    Yun, Z.: The fundamental lemma of Jacquet–Rallis in positive characteristics. Duke Math. J. 156(2), 167–228 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Zhang, W.: Relative trace formula and arithmetic Gross–Prasad conjecture. Preprint (2009). http://www.math.harvard.edu/~wzhang/math/RTF/AFL.pdf
  29. 29.
    Zhang, W.: Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups. Preprint (2011) Google Scholar
  30. 30.
    Zhang, S.: Heights of Heegner points on Shimura curves. Ann. Math. (2) 153(1), 27–147 (2001) zbMATHCrossRefGoogle Scholar
  31. 31.
    Zhang, S.: Gross–Zagier formula for GL 2. Asian J. Math. 5(2), 183–290 (2001) MathSciNetzbMATHGoogle Scholar
  32. 32.
    Zhang, S.: Linear forms, algebraic cycles, and derivatives of L-series. Available at: www.math.columbia.edu/~szhang/papers/Preprints.htm

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations