Skip to main content
Log in

Eisenstein series for SL(2)

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local Whittaker functions over SL2. They are used to compute both the value and derivatives of these functions at critical points.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abramowitz M, Stegun I. Pocketbook of Mathematical Functions. Abridged edition of Handbook of mathematical functions. Material selected by Michael Danos and Johann Rafelski. Thun: Verlag Harri Deutsch, 1984

    MATH  Google Scholar 

  2. Borcherds R E. Automorphic forms with singularities on Grassmannians. Invent Math, 1998, 132: 491–562

    Article  MATH  MathSciNet  Google Scholar 

  3. Bruinier J, Burgos Gil J, Kühn U. Borcherds products and arithmetic intersection theory on Hilbert modular surfaces. Duke Math J, 2007, 139: 1–88

    Article  MATH  MathSciNet  Google Scholar 

  4. Bruinier J H, Funke J. On two geometric theta lifts. Duke Math J, 2004, 125: 45–90

    Article  MATH  MathSciNet  Google Scholar 

  5. Bruinier J, Kudla S, Yang T H. Faltings heights of Big CM cycles and derivatives of L-functions. Preprint, 2010

  6. Bruinier J, Kühn U. Integrals of automorphic Green’s functions associated to Heegner divisors. IMRN, 2003, 31: 1687–1729

    Article  Google Scholar 

  7. Bruinier J, Yang T H. CM-values of Hilbert modular functions. Invent Math, 2006, 163: 229–288

    Article  MATH  MathSciNet  Google Scholar 

  8. Bruinier J, Yang T H. Faltings heights of CM cycles and derivatives of L-functions. Invent Math, 2009, 177: 631–681

    Article  MATH  MathSciNet  Google Scholar 

  9. Bruinier J, Yang T H. CM values of automorphic Green functions on orthogonal groups over totally real fields. Arithmetic Geometry and Automorphic Forms, to appear

  10. Cassels J W S. Rational Quadratic Forms, London Mathematical Society Monographs, 13. London-New York: Academic Press, 1978.

    MATH  Google Scholar 

  11. Cohen H. Sums involving the values at negative integers of L-functions of quadratic characters. Math Ann, 1975, 217: 271–285

    Article  MATH  MathSciNet  Google Scholar 

  12. Errthum E. Singular moduli of Shimura curves. PhD Thesis, University of Maryland, 2007

  13. Gross B, Zagier D. On singular moduli. J Reine Angew Math, 1985, 355: 191–220

    MATH  MathSciNet  Google Scholar 

  14. Gross B, Zagier D. Heegner points and derivatives of L-series. Invent Math, 1986, 84: 225–320

    Article  MATH  MathSciNet  Google Scholar 

  15. Hirzebruch F, Zagier D. Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus. Invent Math, 1976, 36: 57–113

    Article  MATH  MathSciNet  Google Scholar 

  16. Kitaoka Y. Arithmetic of quadratic forms. In: Cambridge Tracts in Mathematics 106. Cambridge: Cambridge University Press, 1993

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Kudla S. Integrals of Borcherds forms. Compositio Math, 2003, 137: 293–349

    Article  MATH  MathSciNet  Google Scholar 

  19. Kudla S. Splitting metaplectic covers of dual reductive pairs. Israel J Math, 1994, 87: 361–401

    Article  MATH  MathSciNet  Google Scholar 

  20. Kudla S, Rallis S. On the Weil-Siegel formula. J Reine Angew Math, 1988, 387: 1–68

    MATH  MathSciNet  Google Scholar 

  21. Kudla S, Rallis S. On the Weil-Siegel formula. II. The isotropic convergent case. J Reine Angew Math, 1988, 391: 65–84

    MATH  MathSciNet  Google Scholar 

  22. Kudla S, Rapoport M, Yang T H. On the derivative of an Eisenstein series of weight one. IMRN, 1999, 7: 347–385

    Article  MathSciNet  Google Scholar 

  23. Kudla S, Rapoport M, Yang T H. Derivatives of Eisenstein series and Faltings heights. Compositio Math, 2004, 140: 887–951

    Article  MATH  MathSciNet  Google Scholar 

  24. Kudla S, Rapoport M, Yang T H. Modular forms and special cycles on Shimura curves. In: Annals of Math Studies Series, vol 161, Princeton University Press, 2006

  25. Kudla S, Yang T H. On the pullback of an arithmetic theta function I. Preprint, 2010

  26. Langlands R. On the Functional Equations Satisfied by Eisenstein Series. In: Lecture Notes in Mathematics, vol. 544. Berlin-New York: Springer-Verlag, 1976

    Google Scholar 

  27. Lebedev N N. Special Functions and Their Applications. New York: Dover, 1972

    MATH  Google Scholar 

  28. Rallis S. On the Howe duality conjecture, Compositio Math, 1984, 51: 333–399

    MATH  MathSciNet  Google Scholar 

  29. Ranga Rao R. On some explicit formulas in the theory of Weil representation. Pacific J Math, 1993, 157: 335–371

    MATH  MathSciNet  Google Scholar 

  30. Schofer J. Borcherds forms and generalizations of singular moduli. J Reine Angew Math, 2009, 629: 1–36

    MATH  MathSciNet  Google Scholar 

  31. Siegel C L. Über die analytische Theorie der quadratischen Formen. II. Ann of Math, 1936, 37: 230–263

    Article  MathSciNet  Google Scholar 

  32. Weil A. Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math, 1965, 113: 1–7

    Article  MATH  MathSciNet  Google Scholar 

  33. Yang T H. An explicit formula for local densities of quadratic forms. J Number Theory, 1998, 72: 309–356

    Article  MATH  MathSciNet  Google Scholar 

  34. Zagier D. Nombres de classes et formes modulaires de poids 3/2. C R Acad Sci Paris Sr A-B, 1975, 281: 883–886

    MATH  MathSciNet  Google Scholar 

  35. Zhao L L. Period integral of automorphic Green functions. PhD Thesis, University of Wisconsin, 2011

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to TongHai Yang.

Additional information

Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kudla, S.S., Yang, T. Eisenstein series for SL(2). Sci. China Math. 53, 2275–2316 (2010). https://doi.org/10.1007/s11425-010-4097-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-4097-1

Keywords

MSC(2000)

Navigation