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Converse theorems for automorphic distributions and Maass forms of level N

  • Tadashi Miyazaki
  • Fumihiro Sato
  • Kazunari SugiyamaEmail author
  • Takahiko Ueno
Research
  • 19 Downloads

Abstract

We investigate the relationship between L-functions satisfying certain functional equations, summation formulas of Ferrar–Suzuki type and Maass forms of integral and half-integral weight. Summation formulas of Ferrar–Suzuki type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of SL(2). Our goal is converse theorems for automorphic distributions and Maass forms of level N characterizing them by analytic properties of the associated L-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms studied by Peter and the fourth author.

Keywords

Maass forms Automorphic distributions Converse theorem Summation formula Prehomogeneous zeta functions 

Notes

References

  1. 1.
    Bruggeman, R., Lewis, J., Zagier, D.: Function theory related to the group \(PSL_2({\mathbb{R}})\). In: From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developmental Mathematics, vol. 28, pp. 107–201. Springer, New York (2013)Google Scholar
  2. 2.
    Diamantis, N., Goldfeld, D.: A converse theorem for double Dirichlet series. Am. J. Math. 133, 913–938 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Diamantis, N., Goldfeld, D.: A converse theorem for double Dirichlet series and Shintani zeta functions. J. Math. Soc. Jpn. 66, 449–477 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Erdelyi, A.: Higher Transcendental Functions, vol. I. MacGraw-Hill, New York (1953)zbMATHGoogle Scholar
  5. 5.
    Erdelyi, A.: Tables of Integral Transforms, vol. I. MacGraw-Hill, New York (1954)zbMATHGoogle Scholar
  6. 6.
    Ferrar, W.L.: Summation formulae and their relation to Dirichlet’s series. Compos. Math. 1, 344–360 (1935)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ferrar, W.L.: Summation formulae and their relation to Dirichlet’s series II. Compos. Math. 4, 394–405 (1937)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gelbart, S., Miller, S.D.: Reimann’s zeta function and beyond. Bull. Am. Math. Soc. 41, 59–112 (2004)zbMATHCrossRefGoogle Scholar
  9. 9.
    Gel’fand, I.M., Shilov, G.E.: Generalized Functions, Properties and Operations, vol. 1. Academic Press, New York (1964)zbMATHGoogle Scholar
  10. 10.
    Hamburger, H.: Über einige Beziehungen, die mit der Funktionalgleichung der Riemannschen \(\zeta \)-Funktion äquivalent sind. Math. Ann. 85, 129–140 (1922)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Igusa, J.: An Introduction to the Theory of Local Zeta Functions. AMS/IP Studies in Advanced Mathematics, vol. 14. American Mathematical Society, Providence (2000)Google Scholar
  12. 12.
    Kato, S.: A remark on Maass wave forms attached to real quadratic fields. J. Fac. Sci. Univ. Tokyo Sect. IA 34, 193–201 (1987)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kimura, T.: Introduction to Prehomogeneous Vector Spaces. Translations of Mathematical Monographs, vol. 215. American Mathematical Society, Providence (2003). Translated by Makoto Nagura and Tsuyoshi Niitani and revised by the authorGoogle Scholar
  14. 14.
    Lee, J.: A functional equation and degenerate principal series. Rocky Mt J. Math. 46, 1987–2016 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lewis, J.: Eigenfunctions on symmetric spaces with distribution-valued boundary forms. J. Funct. Anal. 29, 287–307 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lewis, J.B., Zagier, D.: Period functions for Maass wave forms. I. Ann. Math. 153, 191–258 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Maass, H.: Lectures on Modular Functions of One Complex Variable. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 29. Tata Institute of Fundamental Research, Mumbai (1983)CrossRefGoogle Scholar
  18. 18.
    Miller, S.D., Schmid, W.: Summation formulas, form Poisson and Voronoi to the present. In: Noncommutative Harmonic Analysis: In Honor of Jacques Carmona. Progress in Mathematics, vol. 220. Birkhäuser Boston, Boston (2004)CrossRefGoogle Scholar
  19. 19.
    Miller, S.D., Schmid, W.: Distributions and analytic continuation of Dirichlet series. J. Funct. Anal. 214, 155–220 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Miller, S. D., Schmid, W.: The Rankin-Selberg method for automorphic distributions. In: Representation Theory and Automorphic Forms. Progress in Mathematics, vol. 255, pp. 111–150. Birkhäuser, Boston (2008)Google Scholar
  21. 21.
    Miyake, T.: Modular Forms. Springer Monographs in Mathematics. Springer, Berlin (2006). Translated from the 1976 Japanese original by Yoshitaka MaedaGoogle Scholar
  22. 22.
    Mizuno, Y.: Dirichlet series of two variables, real analytic Jacobi-Eisenstein series of matrix index, and Katok-Sarnak type result. Forum Math. 30, 1437–1460 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Neururer, M., Oliver, T.: Weil’s converse theorem for Maass forms and cancellation of zeros. arXiv:1809.06586v3 (2019)
  24. 24.
    Oshima, T.: Boundary Value Problems of Regular Singularities and Representation Theory. Sophia Kokyuroku in Mathematics, vol. 5. Sophia University, Tokyo (1979) (in Japanese)Google Scholar
  25. 25.
    Oshima, T., Sekiguchi, J.: Eigenspaces of invariant differential operators on an affine symmetric space. Invent. Math. 57, 1–81 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Peter, M.: Dirichlet series in two variables. J. Reine Angew. Math. 522, 27–50 (2000)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Peter, M.: Dirichlet series and automorphic functions associated to a quadratic form. Nagoya Math. J. 171, 1–50 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Razar, M.J.: Modular forms for \(\Gamma _{0}(N)\) and Dirichlet series. Trans. Am. Math. Soc. 231, 489–495 (1977)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Saito, H.: On \(L\) functions associated with the vector space of binary quadratic forms. Nagoya Math. J. 130, 149–176 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Saito, H.: Convergence of the zeta functions of prehomogeneous vector spaces. Nagoya Math. J. 170, 1–31 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Sato, F.: On zeta functions of ternary zero forms. J. Fac. Sci. Univ. Tokyo Sect. IA 28, 585–604 (1981)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Sato, F.: Zeta functions in several variables associated with prehomogeneous vector spaces I: functional equations. Tohoku Math. J. 34, 437–483 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Sato, F.: Zeta functions in several variables associated with prehomogeneous vector spaces II: a convergence criterion. Tohoku Math. J. 35, 77–99 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Sato, F.: On functional equations of zeta distributions. Adv. Stud. Pure Math. 15, 465–508 (1989)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sato, F.: The Hamburger theorem for Epstein zeta functions. In: Algebraic Analysis—Papers Dedicated to Professor Mikio Sato on the Occasion of his Sixtieth Birthday, Vol. 2, pp. 789–807 . Academic Press (1989)Google Scholar
  36. 36.
    Shimeno, N.: Boundary values of the Eisenstein series on \(SL(2,\mathbf{Z})\backslash SL(2,\mathbf{R})\). Bull. Okayama Univ. Sci. 35, 9–14 (1999)Google Scholar
  37. 37.
    Schmid, W.: Automorphic distributions for \(SL_2({\mathbb{R}})\). In: Conférence Mosché Flato 1999. Mathematical Physics Studies, Vol. 1 (Dijon), vol. 21, pp. 345–387. Kluwer, Dordrecht (2000)Google Scholar
  38. 38.
    Shimura, G.: On modular forms of half integral weight. Ann. Math. 97, 440–481 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Shintani, T.: On zeta-functions associated with the vector space of quadratic forms. J. Fac. Sci. Univ. Tokyo Sect. IA 22, 25–65 (1975)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Stark, H.M.: \(L\)-functions and character sums for quadratic forms (I). Acta Arith. XIV, 35–50 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Suzuki, T.: Distributions with automorphy and Dirichlet series. Nagoya Math. J. 73, 157–169 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Suzuki, T.: Weil-type representations and automorphic forms. Nagoya Math. J. 77, 145–166 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Tamura, K.: Zeta functions attached to automorphic pairs of distributions on prehomogeneous vector spaces. Comment. Math. Univ. St. Pauli 66, 63–84 (2017)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Ueno, T.: Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related quadratic forms. Nagoya Math. J. 175, 1–37 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Unterberger, A.: Quantization and Non-holomorphic Modular Forms. Lecture Notes in Mathematics, vol. 1742. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
  46. 46.
    Wallach, N.: Real Reductive Groups II. Academic Press, San Diego (1992)zbMATHGoogle Scholar
  47. 47.
    Weil, A.: Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168, 149–156 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Yoshida, H.: Remarks on metaplectic representations of \(SL(2)\). J. Math. Soc. Jpn. 44, 351–373 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, College of Liberal Arts and SciencesKitasato UniversitySagamiharaJapan
  2. 2.Institute for Mathematics and Computer ScienceTsuda CollegeKodaira-shiJapan
  3. 3.Department of MathematicsChiba Institute of TechnologyNarashinoJapan
  4. 4.Medical InformaticsSt. Marianna University School of MedicineKawasakiJapan

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