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Nonanalytic automorphic integrals on the Hecke groups

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Abstrakty
EN
1. Introduction. Since its genesis over a century ago in work of Jacobi, Riemann, Poincar ́e and Klein [Ja29, Ri53, Le64], the theory of automorphic forms has burgeoned from a branch of analytic number theory into an industry all its own. Natural extensions of the theory are to integrals [Ei57, Kn94a, KS96, Sh94], thereby encompassing Hurwitz’s prototype, the analytic weight 2 Eisenstein series [Hu81], and to nonanalytic forms [He59, Ma64, Sel56, ER74, Fr85]. A generalization in both directions at once has also been the subject of some scrutiny. In the present study, inspired by unpublished work of Knopp [Kn94], we consider the nonanalytic automorphic integral.
Twórcy
  • Department of Mathematical Sciences, Villanova University, Villanova, PA 19085, U.S.A.
Bibliografia
  • [Ap90] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd ed., Springer, New York, 1990.
  • [Ei57] M. Eichler, Eine Verallgemeinerung der Abelsche Integrale, Math. Z. 67 (1957), 267-298.
  • [ER74] J. Elstrodt und W. Roelcke, Über das wesentliche Spektrum zum Eigenwertproblem der automorphen Formen, Manuscripta Math. 11 (1974), 391-406.
  • [Fr85] S. Friedberg, Differential operators and theta series, Trans. Amer. Math. Soc. 287 (1985), 569-589.
  • [Gu63] R. C. Gunning, Lectures on Modular Forms, Ann. of Math. Stud. 48, Princeton Univ. Press, Princeton, 1963.
  • [Ha99] A. Hassen, Log-polynomial period functions for Hecke groups, Ramanujan J. 3 (1999), 119-151.
  • [He38] E. Hecke, Lectures on Dirichlet Series, Modular Functions and Quadratic Forms, Edwards Brothers, Ann Arbor, 1938.
  • [He59] E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959.
  • [Hu81] A. Hurwitz, Grundlagen einer independenten Theorie der elliptischen Modulfunctionen und Theorie der Multiplicatorgleichungen erster Stufe, Math. Ann. 18 (1881), 528-591.
  • [Iw97] H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, Amer. Math. Soc., Providence, 1997.
  • [Ja29] C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum, 1829.
  • [Kn83] M. I. Knopp, The weight-changing operator and the Mellin transform of modular integrals, in: Lecture Notes in Math. 1013, Springer, 1983, 284-291.
  • [Kn86] M. I. Knopp, On the Fourier coefficients of small positive powers of θ(τ), Invent. Math. 85 (1986), 165-183.
  • [Kn89] M. I. Knopp, On the Fourier coefficients of cusp forms having small positive weight, in: Proc. Sympos. Pure Math. 49, Part 2, Amer. Math. Soc., 1989, 111-127.
  • [Kn89a] M. I. Knopp, Recent developments in the theory of rational period functions, in: Lecture Notes in Math. 1383, Springer, 1989, 111-122.
  • [Kn93] M. I. Knopp, Modular Functions in Analytic Number Theory, 2nd ed., Chelsea, New York, 1993.
  • [Kn94] M. I. Knopp, Nonanalytic Modular Integrals, unpublished lectures at Bryn Mawr College, June-July 1994.
  • [Kn94a] M. I. Knopp, On Dirichlet series satisfying Riemann's functional equation, Invent. Math. 117 (1994), 361-372.
  • [KS96] M. I. Knopp and M. Sheingorn, On Dirichlet series and Hecke triangle groups of infinite volume, Acta Arith. 76 (1996), 227-244.
  • [Le64] J. Lehner, Discontinuous Groups and Automorphic Functions, Math. Surveys Monographs 8, Amer. Math. Soc., Providence, 1964.
  • [Le66] J. Lehner, A Short Course in Automorphic Functions, Holt, Rinehart, and Winston, New York, 1966.
  • [Ma38] H. Maass, Konstruktion ganzer Modulformen halbzahliger Dimension mit Theta Multiplikatoren in einer und zwei Variablen, Abh. Math. Sem. Hansische Univ. 12 (1938), 133-162.
  • [Ma64] H. Maass, Lectures on Modular Functions of One Complex Variable, Lectures on Math. 29, Tata Institute of Fundamental Research, Bombay, 1964.
  • [Pa98] P. C. Pasles, Convergence of Poincaré series with two complex coweights, Contemp. Math., to appear.
  • [Pa99] P. C. Pasles, A Hecke correspondence theorem for nonanalytic automorphic integrals on the Hecke groups, submitted.
  • [Ra77] R. Rankin, Modular Functions and Forms, Cambridge Univ. Press, Cambridge, 1977.
  • [Ri53] B. Riemann, Collected Works, 2nd ed., Dover, 1953.
  • [Sc74] B. Schoeneberg, Elliptic Modular Functions: An Introduction, Springer, New York, 1974.
  • [Sel56] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87.
  • [Ser70] J.-P. Serre, A Course in Arithmetic, Springer, New York, 1970.
  • [Sh94] G. Shimura, Introduction to Arithmetic Theory of Automorphic Forms, Publ. Math. Soc. Japan 11, Princeton Univ. Press, Princeton, 1994.
  • [Si56] C. L. Siegel, Die Funktionalgleichungen einiger Dirichletscher Reihen, Math. Z. 63 (1956), 363-373.
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Bibliografia
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