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A refined Poisson summation formula for certain Braverman-Kazhdan spaces

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Abstract

Braverman and Kazhdan (2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands’ functoriality conjecture. As an evidence for their conjectures, Braverman and Kazhdan (2002) considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. The connection between the two papers is made explicit in the work of Li (2018). In this paper, we consider a special case of the setting in Braverman and Kazhdan’s later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold (in a slightly different setting) in the work of Braverman and Kazhdan (2002).

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References

  1. Blomer V, Brumley F. On the Ramanujan conjecture over number fields. Ann of Math (2), 2011, 174: 581–605

    Article  MathSciNet  Google Scholar 

  2. Bouthier A, Ngô B C, Sakellaridis Y. On the formal arc space of a reductive monoid. Amer J Math, 2016, 138: 81–108

    Article  MathSciNet  Google Scholar 

  3. Braverman A, Kazhdan D. Normalized intertwining operators and nilpotent elements in the Langlands dual group. Mosc Math J, 2002, 2: 533–553

    Article  MathSciNet  Google Scholar 

  4. Braverman A, Kazhdan D. γ-functions of representations and lifting. In: Visions in Mathematics. Modern Birkh’user Classics. Basel: Birkh’user, 2010, 237–278

    Google Scholar 

  5. Brumley F. Effective multiplicity one on GLN and narrow zero-free regions for Rankin-Selberg L-functions. Amer J Math, 2006, 128: 1455–1474

    Article  MathSciNet  Google Scholar 

  6. Cheng S, Ngo B C. On a conjecture of Braverman and Kazhdan. Int Math Res Not IMRN, 2018, 20: 6177–6200

    Article  MathSciNet  Google Scholar 

  7. Duke W, Rudnick Z, Sarnak P. Density of integer points on affine homogeneous varieties. Duke Math J, 1993, 71: 143–179

    Article  MathSciNet  Google Scholar 

  8. Folland G B. A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. Boca Raton: CRC Press, 1995

    Google Scholar 

  9. Franke J, Manin Y I, Tschinkel Y. Rational points of bounded height on Fano varieties. Invent Math, 1989, 95: 421–435

    Article  MathSciNet  Google Scholar 

  10. 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  MathSciNet  Google Scholar 

  11. Gelbart S, Piatetski-Shapiro I, Rallis S. Explicit Constructions of Automorphic L-Functions. Lecture Notes in Mathematics, vol. 1254. Berlin: Springer-Verlag, 1987

    Google Scholar 

  12. Gelbart S, Shahidi F. Boundedness of automorphic L-functions in vertical strips. J Amer Math Soc, 2001, 14: 79–107

    Article  MathSciNet  Google Scholar 

  13. Getz J R. Nonabelian Fourier transforms for spherical representations. Pacific J Math, 2018, 294: 351–373

    Article  MathSciNet  Google Scholar 

  14. Getz J R, Liu B. A summation formula for triples of quadratic spaces. Adv Math, 2019, 347: 150–191

    Article  MathSciNet  Google Scholar 

  15. Godement R, Jacquet H. Zeta Functions of Simple Algebras. Lecture Notes in Mathematics, vol. 260. Berlin-New York: Springer-Verlag, 1972

    Google Scholar 

  16. Ikeda T. On the location of poles of the triple L-functions. Compos Math, 1992, 83: 187–237

    MathSciNet  MATH  Google Scholar 

  17. Jacquet H. Archimedean Rankin-Selberg integrals. In: Automorphic forms and L-functions II. Local Aspects. Contemporary Mathematics, vol. 489. Providence: Amer Math Soc, 2009, 57–172

    Google Scholar 

  18. Kudla S S, Rallis S. Poles of Eisenstein series and L-functions. In: Festschrift in honor of II. Israel Mathematical Conference Proceedings, vol. 3. Ramat Gan: Bar-Ilan University, 1990, 81–110

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Lafforgue L. Noyaux du transfert automorphe de Langlands et formules de Poisson non linéaires. Jpn J Math, 2014, 9: 1–68

    Article  MathSciNet  Google Scholar 

  21. Li W-W. Basic functions and unramified local L-factors for split groups. Sci China Math, 2017, 60: 777–812

    Article  MathSciNet  Google Scholar 

  22. Li W-W. Appendix: A comparison of Basic Functions. Geometric Aspects of the Trace Formula. New York: Springer, 2018

    Google Scholar 

  23. Li W-W. Towards Generalized Prehomogeneous Zeta Integrals. Lecture Notes in Mathematics, vol. 2221. Berlin: Springer-Verlag, 2018

    Google Scholar 

  24. Li W-W. Zeta Integrals, Schwartz Spaces and Local Functional Equations. Lecture Notes in Mathematics, vol. 2228. Berlin: Springer-Verlag, 2018

    Google Scholar 

  25. Moreno C J. Advanced Analytic Number Theory: L-Functions. Mathematical Surveys and Monographs, vol. 115. Providence: Amer Math Soc, 2005

    Google Scholar 

  26. Mu’ller W. On the singularities of residual intertwining operators. Geom Funct Anal, 2000, 10: 1118–1170

    Article  MathSciNet  Google Scholar 

  27. Ngô B C. On a certain sum of automorphic L-functions. Contemp Math, 2014, 614: 337–343

    Article  MathSciNet  Google Scholar 

  28. Piatetski-Shapiro I, Rallis S. Rankin triple L functions. Compos Math, 1987, 64: 31–115

    MathSciNet  MATH  Google Scholar 

  29. Sakellaridis Y. Spherical varieties and integral representations of L-functions. Algebra Number Theory, 2012, 6: 611–667

    Article  MathSciNet  Google Scholar 

  30. Sakellaridis Y. Inverse Satake Transforms. Geometric Aspects of the Trace Formula. New York: Springer, 2018

    MATH  Google Scholar 

  31. Serre J-P. Galois Cohomology. Berlin: Springer-Verlag, 1997

    Book  Google Scholar 

  32. Shahidi F. Eisenstein Series and Automorphic L-Functions. American Mathematical Society Colloquium Publications, vol. 58. Providence: Amer Math Soc, 2010

    Google Scholar 

  33. Shahidi F. Local Factors, Reciprocity and Vinberg Monoids. Prime Numbers and Representation Theory. Lecture Series of Modern Number Theory, vol. 2. Beijing: Science Press, 2018

    Google Scholar 

  34. Shahidi F. On generalized Fourier Transforms for Standard L-Functions. Geometric Aspects of the Trace Formula. New York: Springer, 2018

    Google Scholar 

  35. Tate J T. Fourier analysis in number fields and Hecke’s zeta-functions. PhD Thesis. Princeton: Princeton University, 1950

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Acknowledgements

This work was supported by the National Science Foundation of the USA (Grant Nos. DMS-1405708 and DMS-1901883). The second author was supported by the National Science Foundation of the USA (Grant Nos. DMS-1702218 and DMS-1848058), and by a start-up fund from the Department of Mathematics at Purdue University. The authors thank Herve Jacquet and Aaron Pollack for their interest in the results in this paper and for helpful comments and suggestions. The authors thank Wen-Wei Li, Yiannis Sakellaridis, and Freydoon Shahidi for useful conversations, and thank Freydoon Shahidi and Wen-Wei Li for sharing [22, 34] with them. The authors also thank Heekyoung Hahn for the help with editing and for her constant encouragement. Thanks are due to the anonymous referees for several useful comments. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Jayce Robert Getz

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Baiying Liu

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  1. Jayce Robert Getz
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  2. Baiying Liu
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Correspondence to Baiying Liu.

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Getz, J.R., Liu, B. A refined Poisson summation formula for certain Braverman-Kazhdan spaces. Sci. China Math. 64, 1127–1156 (2021). https://doi.org/10.1007/s11425-018-1616-0

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