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Periods and (χ, b)-factors of cuspidal automorphic forms of symplectic groups

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Abstract

In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations σ of symplectic groups Sp2n(A), which detects the right-most pole of the L-function L(s, σ × χ) for some character χ of F×A × of order at most 2, and hence the occurrence of a simple global Arthur parameter (χ, b) in the global Arthur parameter ψ attached to σ. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.

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References

  1. J. G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in G(Q), Duke Mathematical Journal 45 (1978), 911–952.

    Article  MathSciNet  Google Scholar 

  2. J. Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Mathematica 40 (1980), 87–121.

    MathSciNet  MATH  Google Scholar 

  3. J. Arthur, The Endoscopic Classification of Representations, American Mathematical Society Colloquium Publications, Vol. 61, American Mathematical Society, Providence, RI, 2013.

  4. N. Bergeron, J. Millson and C. Moeglin, The hodge conjecture and arithmetic quotients of complex balls, Acta Mathematica 216 (2016), 1–125.

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Bergeron, J. Millson and C. Moeglin, Hodge type theorems for arithmetic manifolds associated to orthogonal groups, International Mathematics Research Notices 15 (2017), 4495–4624.

    MathSciNet  MATH  Google Scholar 

  6. D. Ginzburg, D. Jiang, S. Rallis D. Soudry, L-functions for symplectic groups using Fourier–Jacobi models, in Arithmetic Geometry and Automorphic Forms, Advanced Lectures in Mathematics, Vol. 19, International Press, Somerville, MA, 2011, pp. 183–207.

    Google Scholar 

  7. D. Ginzburg, D. Jiang and D. Soudry, Poles of L-functions and theta liftings for orthogonal groups, Journal of the Institute of Mathematics of Jussieu 8 (2009), 693–741.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Gelbart, I. Piatetski-Shapiro and S. Rallis, Explicit Constructions of Automorphic L-functions, Lecture Notes in Mathematics, Vol. 1254, Springer-Verlag, Berlin, 1987.

  9. W. T. Gan, Y. Qiu and S. Takeda, The regularized Siegel–Weil formula (the second term identity) and the Rallis inner product formula, Inventiones Mathematicae 198 (2014), 739–831.

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Ginzburg, S. Rallis and D. Soudry, The Descent Map from Automorphic Representations of GL(n) to Classical Groups, World Scientific Publishing, Hackensack, NJ, 2011.

    Book  MATH  Google Scholar 

  11. H. He and J. W. Hoffman, Picard groups of Siegel modular 3-folds and θ-liftings, Journal of Lie Theory 22 (2012), 769–801.

    MathSciNet  MATH  Google Scholar 

  12. R. Howe, Automorphic forms of low rank, in Noncommutative Harmonic Analysis and Lie Groups (Marseille, 1980), Lecture Notes in Mathematics, Vol. 880, Springer, Berlin–New York, 1981, pp. 211–248.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Ichino, On the regularized Siegel–Weil formula, Journal für die Reine und Angewandte Mathematik 539 (2001), 201–234.

    MathSciNet  MATH  Google Scholar 

  14. T. Ikeda, On the location of poles of the triple L-functions, Compositio Mathematica 83 (1992), 187–237.

    MathSciNet  MATH  Google Scholar 

  15. T. Ikeda, On the residue of the Eisenstein series and the Siegel–Weil fomula, Compositio Mathematica 103 (1996), 183–218.

    MathSciNet  MATH  Google Scholar 

  16. D. Jiang, Automorphic integral transforms for classical groups I: Endoscopy correspondences, in Automorphic Forms and Related Geometry: Assessing the Legacy of I. I. Piatetski-Shapiro, Contemporary Mathematics, Vol. 614, American Mathematical Society, Providence, RI, 2014, pp. 179–242.

    Chapter  Google Scholar 

  17. H. Jacquet and S. Rallis, Symplectic periods, Journal für die Reine und Angewandte Mathematik 423 (1992), 175–197.

    MathSciNet  MATH  Google Scholar 

  18. D. Jiang and C. Wu, On (χ, b)-factors of cuspidal automorphic representations of unitary groups. II, preprint, 2014.

    Google Scholar 

  19. D. Jiang and C. Wu, On (χ, b)-factors of cuspidal automorphic representations of unitary groups I, Journal of Number Theory 161 (2016), 88–118.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. S. Kudla and S. Rallis, Poles of Eisenstein series and L-functions, in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Part II (Ramat Aviv, 1989), Israel Mathematical Conference Proceedings, Vol. 3, Weizmann, Jerusalem, 1990, pp. 81–110.

    MathSciNet  MATH  Google Scholar 

  21. S. S. Kudla and S. Rallis, A regularized Siegel–Weil formula: the first term identity, Annals of Mathematics 140 (1994), 1–80.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. P. Langlands, Euler Products, Yale Mathematical Monographs, Vol. 1, Yale University Press, New Haven, Conn.-London, 1971.

  23. J.-S. Li, Nonexistence of singular cusp forms, Compositio Mathematica 83 (1992), 43–51.

    MathSciNet  Google Scholar 

  24. C.ˆ Moeglin, Non nullité de certains relevements par séries théta, Journal of Lie Theory 7 (1997), 201–229.

    MathSciNet  MATH  Google Scholar 

  25. C. Moeglin, Quelques propriétés de base des séries théta, Journal of Lie Theory 7 (1997), 231–238.

    MathSciNet  MATH  Google Scholar 

  26. S. D. Miller and W. Schmid, On the rapid decay of cuspidal automorphic forms, Advances in Mathematics 231 (2012), 940–964.

    Article  MathSciNet  MATH  Google Scholar 

  27. C. Moeglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics, Vol. 113, Cambridge University Press, Cambridge, 1995.

  28. I. Piatetski-Shapiro and S. Rallis, A new way to get Euler products, Journal für die Reine und Angewandte Mathematik 392 (1998), 110–124.

    MathSciNet  MATH  Google Scholar 

  29. F. Shahidi, Eisenstein Series and Automorphic L-functions, American Mathematical Society Colloquium Publications, Vol. 58, American Mathematical Society, Providence, RI, 2010.

  30. X. Shen, Unramified computation of tensor L-functions on symplectic groups, Ph.D. Thesis, University of Minnesota, ProQuest LLC, Ann Arbor, MI, 2013.

    Google Scholar 

  31. A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Mathematica 111 (1964), 143–211.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Mathematica 113 (1965), 1–87.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Yamana, L-functions and theta correspondence for classical groups, Inventiones Mathematicae 196 (2014), 651–732.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

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

    Dihua Jiang

  2. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Chenyan Wu

Authors
  1. Dihua Jiang
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  2. Chenyan Wu
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Correspondence to Dihua Jiang.

Additional information

The research of the first-named author is supported in part by the NSF Grants DMS–1301567 and DMS–1600685.

The research of the second-named author is supported in part by National Natural Science Foundation of China (#11601087) and by Program of Shanghai Academic/Technology Research Leader (#16XD1400400).

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Jiang, D., Wu, C. Periods and (χ, b)-factors of cuspidal automorphic forms of symplectic groups. Isr. J. Math. 225, 267–320 (2018). https://doi.org/10.1007/s11856-018-1658-4

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  • DOI: https://doi.org/10.1007/s11856-018-1658-4

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