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A Trace Formula of Special Values of Automorphic L-Functions

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Multiple Dirichlet Series, L-functions and Automorphic Forms

Part of the book series: Progress in Mathematics ((PM,volume 300))

Abstract

Deligne introduced the concept of special values of automorphic L-functions. The arithmetic properties of these L-functions play a fundamental role in modern number theory. In this chapter, we prove a trace formula which relates special values of the Hecke, Rankin, and the central value of the Garrett triple L-function attached to primitive new forms. This type of trace formula is new and involves special values in the convergent and nonconvergent domain of the underlying L-functions.

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References

  1. T. Arakawa: Jacobi Eisenstein series and a basis problem for Jacobi forms. Comm. Mathematici Universitatis Sancti Pauli. 43 (1994), 181–216

    MathSciNet  MATH  Google Scholar 

  2. T. Arakawa, B. Heim: Real analytic Jacobi Eisenstein series and Dirichlet series attached to three Jacobi forms. Max-Planck-Institut Bonn. Series 66 (1998)

    Google Scholar 

  3. P. Deligne: Formes modulaires et representations l-adic. Lect. Notes Math. 179 (1971), Berlin-Heidelberg-New York, 139–172.

    Google Scholar 

  4. P. Deligne: Valeurs de fonctions L et periode d’integrales. Proc. Symposia Pure Math. 33 (1979), part 2, 313–346.

    Google Scholar 

  5. T. Dokchitser: Computing special values of motivic L-functions. Exp. Math. 13 (2004), 137–149.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Eichler, D. Zagier: The theory of Jacobi forms. Progress in Mathematics. Vol. 55. Boston-Basel-Stuttgart: Birkhäuser (1985).

    Google Scholar 

  7. P. Garrett: Pullbacks of Eisenstein series; applications. Automorphic forms of several variables (Katata, 1983), 114–137, Progr. Math., 46 Birkhäuser Boston, Boston, MA, 1984.

    Google Scholar 

  8. P. Garrett: Decomposition of Eisenstein series: triple product L-functions. Ann. Math. 125 (1987), 209–235.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. van der Geer: Siegel modular forms. manuscript on arXiv:math.AG/0605346 v1

    Google Scholar 

  10. B. Heim: Über Poincare Reihen und Restriktionsabbildungen. Abh. Math. Sem. Univ. Hamburg. 68 (1998), 79–89

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Heim: Pullbacks of Eisenstein series, Hecke-Jacobi theory and automorphic L-functions. In: Automorphic Froms, Automorphic Representations and Arithmetic. Proceedings of Symposia of Pure Mathematics 66, part 2 (1999).

    Google Scholar 

  12. B. Heim: L-functions for Jacobi forms and the basis problem. Manuscripta Math. 106 (2001), 489–503.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Heim: On injectivity of the Satoh lifting of modular forms and the Taylor coefficients of Jacobi forms.Preprint MPI-Bonn Series 68, 2005

    Google Scholar 

  14. A. Ichino: Pullbacks of Saito-Kurokawa Lifts. Invent. Math. 162 (2005), 551–647.

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Ikeda: On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n. Ann. of Math. 154 no. 3 (2001), 641–681.

    Google Scholar 

  16. T. Ikeda: Pullback of the lifting of elliptic cusp forms and Miyawakis conjecture. Duke Math. Journal, 131 no. 3 (2006), 469–497.

    Google Scholar 

  17. H. Iwaniec: Topics in classical automorphic forms. Vol. 17 Graduate Studies in mathematics, AMS (1997).

    Google Scholar 

  18. H. Iwaniec and P. Sarnak: Perspectives on the analytic theory of L-functions. GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. (2000), Special Volume, Part II, 705–741.

    Google Scholar 

  19. H. Katsurada, Hisa-aki Kawamura: A certain Dirichlet series of Rankin-Selberg type associated with the Ikeda lifting. Department of Mathemtics Hokkaido University, series 2006, number 806

    Google Scholar 

  20. H. Klingen: Über die Werte der Dedekindschen Zetafunktion. Math. Ann. 145 (1962), 265–272.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Klingen: Introductory lectures on Siegel modular forms. Cambridge Studies in Advanced Mathematics, 20. Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  22. W. Kohnen: Modular forms of half-integral weight on Γ 0 (4). Math. Ann. 248 (1980), 249–266.

    Article  MathSciNet  MATH  Google Scholar 

  23. W. Kohnen, N.-P. Skoruppa:A certain Dirichlet series attached to Siegel modular forms of degree two. Invent. Math. 95 (1989), 449–476

    Article  MathSciNet  Google Scholar 

  24. H. Maass: Über eine Spezialschar von Modulformen zweiten Grades I,II,III. Invent. Math. 52, 53, 53 (1979), 95–104, 249–253, 255–265.

    Google Scholar 

  25. S. Mizumoto: Nearly Holomorphic Eisenstein Liftings. Abh. Math. Sem. Univ. Hamburg 67 (1997), 173–194,

    Article  MathSciNet  MATH  Google Scholar 

  26. T. Orloff: Special values and mixed weight triple products (with an appendix by Don Blasius). Invent. Math. 90 (1987).

    Google Scholar 

  27. G. Shimura: On modular forms of half-integral weight. Ann. of Math. 97 (1973), 440–481.

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Shimura: The special values of the zeta functions associated with cusp forms. Comm. pure appl. Math. 29 (1976), 783–804.

    Article  MathSciNet  MATH  Google Scholar 

  29. G. Shimura: Eisenstein series and zeta functions on symplectic groups. Inv. Math. 119 (1995), 539–584.

    Article  MathSciNet  MATH  Google Scholar 

  30. C.L. Siegel: Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Göttingen (1969), 87–102.

    Google Scholar 

  31. D. Zagier: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), pp. 105–169. Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977.

    Google Scholar 

  32. D. Zagier: Sur la conjecture de Saito-Kurokawa (d’après H. Maass). Sém. Delange-Pisot-Poitou 1979/1980, Progress in Math. 12 (1980), 371–394

    Google Scholar 

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Authors and Affiliations

  1. German University of Technology (GUtech), 1816, Athaibah PC 130, Muscat, Sultanate of Oman

    Bernhard Heim

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  1. Bernhard Heim
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Correspondence to Bernhard Heim .

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Editors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, 94305-2125, California, USA

    Daniel Bump

  2. , Department of Mathematics, Boston College, Commonwealth Ave. 140, Chestnut Hill, 02467-3806, Massachusetts, USA

    Solomon Friedberg

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

    Dorian Goldfeld

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Heim, B. (2012). A Trace Formula of Special Values of Automorphic L-Functions. In: Bump, D., Friedberg, S., Goldfeld, D. (eds) Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol 300. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8334-4_8

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