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Periods of automorphic forms, poles of L-functions and functorial lifting

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Abstract

In this paper, we survey our work on period, poles or special values of certain automorphic L-functions and their relations to certain types of Langlands functorial transfers. We outline proofs for some results and leave others as conjectures.

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References

  1. Ben-Artzi A, Soudry D. L-functions for U m × R E/F GL n \( (n \leqslant [\frac{m} {2}]) \). In: Automorphic Forms and L-functions, I: Global Aspects. Israel Math Proc. Contemporary Math, vol. 488. Providence, RI: Amer Math Soc, 2009, 13–61

    Google Scholar 

  2. Borel A, Jacquet H. Automorphic forms and automorphic representations. In: Proc Sympos Pure Math, vol. 33. Automorphic Forms, Representations and L-functions, vol. 1. Providence, RI: Amer Math Soc, 1979, 189–207

    Google Scholar 

  3. Bump D. The Rankin-Selberg method: an introduction and survey. In: Automorphic Representations, L-functions and Applications: Progress and Prospects. Ohio State Univ Math Res Inst Publ, vol. 11. Berlin: de Gruyter, 2005, 41–73

    Google Scholar 

  4. Bump D, Friedberg S. The exterior square automorphic L-functions on GL(n). In: Festschrift in honor of I. I. Piatetski- Shapiro on the occasion of his sixtieth birthday, vol. II. Israel Math Conf Proc, vol. 3. Jerusalem: Weizmann, 1990, 47–65

    Google Scholar 

  5. Bump D, Friedberg S, Ginzburg D. A Rankin-Selberg integral using the automorphic minimal representation of SO(7). J Ramanujan Soc, 2000, 15: 81–124

    MATH  MathSciNet  Google Scholar 

  6. Bump D, Friedberg S, Ginzburg D. Small representations for odd orthogonal groups. Int Math Res Not, 2003, 25: 1363–1393

    Article  MathSciNet  Google Scholar 

  7. Bump D, Ginzburg D. Symmetric square L-functions on GL(r). Ann of Math (2), 1992, 136: 137–205

    Article  MathSciNet  Google Scholar 

  8. Bump D, Ginzburg D. Spin L-functions on symplectic groups. Int Math Res Not, 1992, 8: 153–160

    Article  MathSciNet  Google Scholar 

  9. Bump D, Ginzburg D, Hoffstein J. The symmetric cube. Invent Math, 1996, 125: 413–449

    Article  MATH  MathSciNet  Google Scholar 

  10. Cogdell J, Kim H, Murty M. Lectures on automorphic L-functions. In: Fields Institute Monographs, vol. 20. Providence, RI: Amer Math Soc, 2004

    Google Scholar 

  11. Cogdell J W, Kim H H, Piatetski-Shapiro I I, et al. On lifting from classical groups to GLN. Publ Math Inst Hautes Études Sci, 2001, 93: 5–30

    MATH  MathSciNet  Google Scholar 

  12. Cogdell J W, Kim H H, Piatetski-Shapiro I I, et al. Functoriality for the classical groups. Publ Math Inst Hautes Études Sci, 2004, 99: 163–233

    MATH  MathSciNet  Google Scholar 

  13. Cogdell J, Piatetski-Shapiro I. Remarks on Rankin-Selberg convolutions. In: Contributions to Automorphic Forms, Geometry, and Number Theory. Baltimore: Johns Hopkins Univ Press, 2004, 255–278

    Google Scholar 

  14. Deligne P. Valeurs de fonctions L et pÉriodes d’intÉgrales (in French). In: Proc Sympos Pure Math, vol. 33. Automorphic Forms, Representations and L-Functions, vol. 2. Providence, RI: Amer Math Soc, 1979, 313–346

    Google Scholar 

  15. Franke J. Harmonic analysis in weighted L 2-spaces. Ann Sci Cole Norm Sup, 1998, 31: 181–279

    MATH  MathSciNet  Google Scholar 

  16. Gan W T, Gross B, Prasad D. Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. arXiv: 0909.2999

  17. Gan W T, Takeda S. The local Langlands conjecture for GSp(4). arXiv: 0706.0952

  18. Gan W T, Takeda S. The local Langlands conjecture for Sp(4). arXiv: 0805.2731

  19. Gelbart S, Lapid E M. Lower bounds for L-functions at the edge of the critical strip. Amer J Math, 2006, 128: 619–638

    Article  MATH  MathSciNet  Google Scholar 

  20. Gelbart S, Miller S D. Riemann’s zeta function and beyond. Bull Amer Math Soc, 2004, 41: 59–112

    Article  MATH  MathSciNet  Google Scholar 

  21. Gelbart S, Shahidi F. Analytic properties of automorphic L-functions. In: Perspectives in Mathematics, vol. 6. Boston: Academic Press, 1988

    Google Scholar 

  22. Gilkey P, Seitz G. Some representations of exceptional Lie algebras. In: Geometries and Groups. Geom Dedicata, 1988, 25: 407–416

    MATH  MathSciNet  Google Scholar 

  23. Ginzburg D. On spin L-functions for orthogonal groups. Duke Math J, 1995, 77: 753–798

    Article  MATH  MathSciNet  Google Scholar 

  24. Ginzburg D. On standard L-functions for E6 and E7. J Reine Angew Math, 1995, 465: 101–131

    MATH  MathSciNet  Google Scholar 

  25. Ginzburg D. Endoscopic lifting in classical groups and poles of tensor L-functions. Duke Math J, 2008, 141: 447–503

    Article  MATH  MathSciNet  Google Scholar 

  26. Ginzburg D, Hundley J. Spin L-functions for GSO 10. J Reine Angew Math, 2007, 603: 183–213

    MATH  MathSciNet  Google Scholar 

  27. Ginzburg D, Jiang D. Periods and liftings: from G2 to C3. Israel J Math, 2001, 123: 29–59

    Article  MATH  MathSciNet  Google Scholar 

  28. Ginzburg D, Jiang D, Rallis S. Nonvanishing of the central critical value of the third symmetric power L-functions. Forum Math, 2001, 13: 109–132

    Article  MATH  MathSciNet  Google Scholar 

  29. Ginzburg D, Jiang D, Rallis S. On the nonvanishing of the central value of the Rankin-Selberg L-functions. J Amer Math Soc, 2004, 17: 679–722

    Article  MATH  MathSciNet  Google Scholar 

  30. Ginzburg D, Jiang D, Rallis S. Periods of residual representations of SO(2l). Manuscripta Math, 2004, 113: 319–358

    Article  MATH  MathSciNet  Google Scholar 

  31. 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. Ohio State Univ Math Res Inst Publ, vol. 11. Berlin: de Gruyter, 2005, 157–191

    Google Scholar 

  32. Ginzburg D, Jiang D, Rallis S. Models for certain residual representations of unitary groups. In: The AMS series Contemporary Mathematics for Stephen Gelbart,vol. 488. Providence, RI: Amer Math Soc, 2009, 125–146

    Google Scholar 

  33. Ginzburg D, Jiang D, Rallis S, et al. L-functions for symplectic groups using Fourier-Jacobi models. Preprint, 2009

  34. Ginzburg D, Piatetski-Shapiro I, Rallis S. L Functions for the Orthogonal Group. In: Mem Amer Math Soc, vol. 128. Providence, RI: Amer Math Soc, 1997

    Google Scholar 

  35. Ginzburg D, Rallis S. The exterior cube L-function for GL(6). Compos Math, 2000, 123: 243–272

    Article  MATH  MathSciNet  Google Scholar 

  36. Ginzburg D, Rallis S, Soudry D. L-functions for symplectic groups. Bull Soc Math France, 1998, 126: 181–244

    MATH  MathSciNet  Google Scholar 

  37. Ginzburg D, Rallis S, Soudry D. On explicit lifts of cusp forms from GLm to classical groups. Ann of Math (2), 1999, 150: 807–866

    Article  MATH  MathSciNet  Google Scholar 

  38. Ginzburg D, Rallis S, Soudry D. On a correspondence between cuspidal representations of GL2n and \( \widetilde{Sp}_{2_n } \). J Amer Math Soc, 1999, 12: 849–907

    Article  MATH  MathSciNet  Google Scholar 

  39. Ginzburg D, Rallis S, Soudry D. Lifting cusp forms on GL2n to \( \widetilde{Sp}_{2_n } \): the unramified correspondence. Duke Math J, 1999, 100: 243–266

    Article  MATH  MathSciNet  Google Scholar 

  40. Ginzburg D, Rallis S, Soudry D. Generic automorphic forms on SO(2n + 1): functorial lift to GL(2n), endoscopy, and base change. Int Math Res Not, 2001, 14: 729–764

    Article  MathSciNet  Google Scholar 

  41. Gross B, Prasad D. On the decomposition of a representation of SOn when restricted to SOn-1. Canad J Math, 1992, 44: 974–1002

    MATH  MathSciNet  Google Scholar 

  42. Gross B, Prasad D. On irreducible representations of SO2n+1 × SO2m . Canad J Math, 1994, 46: 930–950

    MATH  MathSciNet  Google Scholar 

  43. Harder G. Modular symbols and special values of automorphic L-functions. Preprint, 2007

  44. Harris M. Cohomological automorphic forms on unitary groups, II: Period relations and values of L-functions. In: Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory. Lect Notes Ser Inst Math Sci Natl Univ Singap, vol. 12. Singapore: World Sci Publ, 2007, 89–149

    Chapter  Google Scholar 

  45. Harris M, Taylor R. The geometry and cohomology of some simple Shimura varieties. In: Ann Math Stud, vol. 151. Princeton: Princeton University Press, 2001

    Google Scholar 

  46. Henniart G. Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique (in French). Invent Math, 2000, 139: 439–455

    Article  MATH  MathSciNet  Google Scholar 

  47. Ichino I, Ikeda T. On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geom Funct Anal, 2010, 19: 1378–1425

    Article  MATH  MathSciNet  Google Scholar 

  48. Ikeda T. On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series. J Math Kyoto Univ, 1994, 34: 615–636

    MATH  MathSciNet  Google Scholar 

  49. Jacquet H. Archimedean Rankin-Selberg integrals. In: Automorphic Forms and L-functions II: Local Aspects. Proceedings of a workshop in honor of Steve Gelbart on the occasion of his 60th birthday. Contemp Math, vol. 489. Providence, RI: Amer Math Soc, 2009, 57–172

    Google Scholar 

  50. Jacquet H, Langlands R. Automorphic forms on GL(2). In: Lecture Notes in Mathematics, vol. 114. Berlin-New York: Springer-Verlag, 1970

    Google Scholar 

  51. Jacquet H, Piatetskii-Shapiro I I, Shalika J A. Rankin-Selberg convolutions. Amer J Math, 1983, 105: 367–464

    Article  MATH  MathSciNet  Google Scholar 

  52. Jacquet H, Rallis S. Symplectic periods. J Reine Angew Math, 1992, 423: 175–197

    MathSciNet  Google Scholar 

  53. Jacquet H, Rallis S. Uniqueness of linear periods. Compos Math, 1996, 102: 65–123

    MATH  MathSciNet  Google Scholar 

  54. Jacquet H, Shalika J. Exterior square L-functions. In: Automorphic Forms, Shimura Varieties, and L-functions, vol. II. Perspect Math, vol. 11. Boston: Academic Press, 1990, 143–226

    Google Scholar 

  55. Jiang D. G 2-periods and residual representations. J Reine Angew Math, 1998, 497: 17–46

    MATH  MathSciNet  Google Scholar 

  56. Jiang D. Periods of automorphic forms. In: Proceedings of the International Conference on Complex Geometry and Related Fields. Stud Adv Math,2007, 39: 125–148

    Google Scholar 

  57. Jiang D, Qin Y J. Residues of Eisenstein series and generalized Shalika models for SO(4n). J Ramanujan Math Soc, 2007, 22: 1–33

    MathSciNet  Google Scholar 

  58. Jiang D, Soudry D. The local converse theorem for SO2n+1 and applications. Ann of Math (2), 2003, 157: 743–806

    Article  MATH  MathSciNet  Google Scholar 

  59. Jiang D, Soudry D. Generic Representations and the local Langlands reciprocity law for p-adic SO(2n + 1). In: Contributions to Automorphic Forms, Geometry, and Number Theory. Baltimore, MD: Johns Hopkins Univ Press, (2004), 457–519

    Google Scholar 

  60. Kudla S S, Rallis S. A regularized Siegel-Weil formula: the first term identity. Ann of Math (2), 1994, 140: 1–80

    Article  MATH  MathSciNet  Google Scholar 

  61. Langlands R. Euler products. In: James K Whittemore Lecture in Mathematics given at Yale University. Yale Mathematical Monographs, vol. 1. New Haven-London: Yale University Press, 1971

    Google Scholar 

  62. Langlands R. On the functional equations satisfied by Eisenstein series. In: Lecture Notes in Mathematics, vol. 544. Berlin: Springer-Verlag, 1976

    Google Scholar 

  63. Langlands R. On the notion of an automorphic representation. In: Proc Sympos Pure Math, vol. 33. Providence, RI: Amer Math Soc, 1979, 189–207

    Google Scholar 

  64. Langlands R P. On the classification of irreducible representations of real algebraic groups. In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups. Math Surveys Monogr, vol. 31. Providence, RI: Amer Math Soc, 1989, 101–170

    Google Scholar 

  65. Liu B Y. Genericity of representations of p-adic Sp2n and local Langlands parameters. Canadian J Math, 2009

  66. Manin Y I. Iterated integrals of modular forms and noncommutative modular symbols. In: Algebraic Geometry and Number Theory. Progr Math, vol. 253. Boston: Birkhäuser Boston, 2006, 565–597

    Chapter  Google Scholar 

  67. Moeglin C, Waldspurger J L. Le spectre rÉsiduel de GL(n) (in French). Ann Sci École Norm Sup, 1989, 22: 605–674

    MATH  MathSciNet  Google Scholar 

  68. Moeglin C, Waldspurger J L. Spectral decomposition and Eisenstein series. In: Cambridge Tracts in Mathematics, vol. 113. Cambridge: Cambridge University Press, 1995

    Google Scholar 

  69. Novodvorsky M. Fonction J pour des groupes orthogonaux. (in French). C R Acad Sci Paris Ser A, 1975, 280: 1421–1422

    MATH  MathSciNet  Google Scholar 

  70. Piatetski-Shapiro I. Work of Waldspurger. In: Lie Group Representations, vol. II. Lecture Notes in Math, vol. 1041. Berlin: Springer-Verlag, 1984, 280–302

    Google Scholar 

  71. Sakellaridis Y. Spherical varieties and integral representations of L-functions. arXiv: 0905.4245

  72. Savin G. Local Shimura correspondence. Math Ann, 1988, 280: 185–190

    Article  MATH  MathSciNet  Google Scholar 

  73. Shahidi F. On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann of Math (2), 1988, 127: 547–584

    Article  MathSciNet  Google Scholar 

  74. Shahidi F. A proof of Langlands’ conjecture on Plancherel measures; Complementary series for p-adic groups. Ann of Math (2), 1990, 132: 273–330

    Article  MathSciNet  Google Scholar 

  75. Soudry D. Rankin-Selberg Convolutions for SO2l+1 × GLn: Local Theory. In: Mem Amer Math Soc, vol. 105. Providence, RI: Amer Math Soc, 1993

    Google Scholar 

  76. Soudry D. On Langlands functoriality from classical groups to GLn, I: Automorphic forms. Asterisque, 2005, 298: 335–390

    MathSciNet  Google Scholar 

  77. Soudry D. Rankin-Selberg integrals, the descent method, and Langlands functoriality. In: International Congress of Mathematicians, vol. II. Zürich: Eur Math Soc, 2006, 1311–1325

    Google Scholar 

  78. Vo S. The spin L-function on the symplectic group GSp(6). Israel J Math, 1997, 101: 1–71

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, 69978, Tel-Aviv, Israel

    David Ginzburg & David Soudry

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    DiHua Jiang

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  1. David Ginzburg
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  2. DiHua Jiang
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  3. David Soudry
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Correspondence to DiHua Jiang.

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

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Ginzburg, D., Jiang, D. & Soudry, D. Periods of automorphic forms, poles of L-functions and functorial lifting. Sci. China Math. 53, 2215–2238 (2010). https://doi.org/10.1007/s11425-010-4020-9

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