Article ID: MTJPAM-D-19-00009

Title: JACOBI FORMS IN TWO VARIABLES: MULTIPLE ELLIPTIC DEDEKIND SUMS, THE KUMMER-VON STAUDT CLAUSEN CONGRUENCES FOR ELLIPTIC BERNOULLI FUNCTIONS AND VALUES OF HECKE L-FUNCTIONS


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-19-00009; Volume 1 / Issue 2 / Year 2019, Pages 58-129

Document Type: Research Paper

Author(s): Abdelmejid Bayad a

aUniversité Paris-Saclay, Laboratoire de Mathématiques et Modélisation d’Évry, CNRS (UMR 8071), 23 Boulevard de France, 91037 Evry cedex, France

Received: 5 December 2019, Accepted: 12 December 2019, Available online: 28 December 2019.

Corresponding Author: Abdelmejid Bayad (Email address: abdelmejid.bayad@univ-evry.fr)

Full Text: PDF

Abstract

In this paper we introduce elliptic Bernoulli functions and numbers, which are related to special Jacobi forms of two variables and study their properties.
More importantly, we state and prove elliptic analogues to the following important theorems:

  1. The Dedekind reciprocity law for Dedekind classical sums, here we introduce enhanced multiple elliptic Dedekind sums and study their reciprocity law.
  2. The congruence of Clausen-von-Staudt and Kummer for Bernoulli numbers, here we state and prove it for elliptic Bernoulli numbers.
  3. We obtain Damerell’s type result concerning the algebraicity of the special values of the Hecke L-function related to our Jacobi forms.
  4. As a corollary, we connect these elliptic Bernoulli numbers ( explicitly computed ) to the special values of Hecke L-functions of imaginary quadratic number field and associated to some Grössencharacter.

Keywords: Elliptic Bernoulli functions, Elliptic Bernoulli numbers, Jacobi forms, Multiple elliptic Dedekind sums, The Kummer-Von Staudt Clausen congruences, Hecke L-function, Weierstrass functions, Eisenstein Series

References:
  1. T. M Apostol, Theorems on generalized Dedekind Sums, Pacific. J. Math, 2, 1-9, 1952.
  2. M.F. Atiyah, The Logarithm of the Dedekind η-Function, Math. Ann, 278, 335-380, 1987.
  3. M.F. Atiyah, F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math.Soc, 65, 276-281, 1959.
  4. J. Barge, E. Ghys,Cocycles d’Euler et de Maslov, Math. Ann. 294 (2), 235-265, 1992.
  5. A. Bayad, Sommes de Dedekind elliptiques et formes de Jacobi, Ann. Instit. Fourier, Vol. 51, Fasc. 1, 29-42, 2001.
  6. A. Bayad, Valuation p-adique et relation de distribution additive pour certaines fonctions q-périodiques, Journal of Number Theory Vol. 65 (1), 1-22,1997.
  7. A. Bayad, Formes de Jacobi et formules de Weber p-adiques, Journal de Théorie des nombres de Bordeaux No11, 317-329, 1999.
  8. A. Bayad, Sommes elliptiques multiples d’Apostol-Dedekind-Zagier, C.R.A.S Paris, Ser. I 339, fascicule 7, Série I, 457-462, 2004.
  9. A. Bayad, Applications aux sommes elliptiques multiples d’Apostol-Dedekind-Zagier, C.R.A.S Paris, Ser. I 339, fascicule 8, Série I, 529-532, 2004.
  10. A. Bayad, E.J Gomez-Ayala, Formes de Jacobi et formules de distribution, Journal of Number theory 109, 136-162, 2004.
  11. A. Bayad, G. Robert, Note sur une forme de Jacobi méromorphe, C.R.A.S Paris, 325,455-460, 1997.
  12. A. Bayad, G. Robert, Amélioration d’une congruence pour certaines éléments de Stickelberger quadratiques, Bulletin de la société mathématique de france, No. 125, 249-267, 1997.
  13. M. BECK, Dedekind cotangent sums, Acta Arithmetica 109 (2), 109-130, 2003.
  14. B.C. BERNDT, Reciprocity theorems for Dedekind sums and generalizations, Adv. in Math. 23 (3), 285-316, 1977.
  15. B.C Berndt,U. Dieter,Sums involving the greatest integer function and Riemann-Stieltjes integration, J. Reine Angew. Math. 337, 208-220, 1982.
  16. L.Carlitz, Arithmetics properties of generalized Bernoulli numbers, J. Reine Angew. Math. 202, 164-182, 1959.
  17. L.Carlitz, Some theorems on generalized Dedekind sums, Pacific J. Math. 3, 513-522, 1953.
  18. T. Clausen, Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen, Astr. Nachr. 17, 351–352, 1840.
  19. U. DIETER, Cotangent sums, a further generalization of Dedekind sums, J. Number Th. 18, 289-305, 1984.
  20. Maria Immaculada Gàlvez Carrillo,Modular invariants for manifolds with Boundary, Thesis, 2001. http://www.tdx.cesca.es/TDX-0806101-095056/
  21. C. Goldstein, N. Schappacher, Séries d’Eisenstein et fonctions L de courbesvelliptiques à multiplication complexe, J. Reine Angew. Math. 327, 184-218, 1981.
  22. E. Grosswald, H. Rademacher, Dedekind Sums, Carus Mathematical Monographs No.16, Mathematical assoc. America, Washington D.C, 1972.
  23. R. R. HALL, J. C. WILSON, D. ZAGIER, Reciprocity formulae for general Dedekind-Rademacher sums, Acta Arith. 73 (4), 389-396, 1995.
  24. H. Hasse, Neue Bergründung der komplexen Multiplikation I, J. Reine Angew. Math, 157, 115-139, 1927.
  25. , G. Herglotz, Uber die Entwicklungskoeffizienten der Weierstrassschen ℘-Funktion, Sitzung, Berichte, Leibzig, 269-289, 1922.
  26. F. Hirzebruch, The signature theorem: reminiscences and recreation , Prospects in Mathematics. Ann. of Math.Studies 70, 3-31, Princeton University Press, Princeton, 1971.
  27. F. Hirzebruch, T. Berger, R. Jung, Manifolds and Modular forms , Aspects of Math.E. 20, Vieweg, 1992.
  28. F. Hirzebruch, D. Zagier, The Atiyah-Singer Theorem and Elementary Number Theory, Math. Lecture Series 3, Publish or Perish Inc, 1974.
  29. A. Hurwitz, Uber die Rntwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann. 51 , 196-226, 1899.
  30. H. Ito, A function on the upper half space which is analogous to the imaginary part of logη(z), J. Reine Angew. Math. 373, 148-165, 1986.
  31. N. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. of Math. 104, 459-571, 1976.
  32. N. Katz, The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers, Math. Ann. 216, 1-4, 1975.
  33. R. Kirby, P. Melvin, Dedekind sums, μ-invariants and the signature cocycle, Math. Annalen 299, 231-267, 1994.
  34. L. Kronecker, Zur Theorie der elliptischen Modulfunktionen, Werke, 4, pp.347-495 and, pp.1–132, Leibzig, 1929.
  35. E. Kummer, Über eine allgemeine Eigenschaft der rationalen Entwicklungskoeffizienten einer bestimmten Gattung analytischer Funktionen, J. Reine Angew. Math. 41, pp.368-372, 1851.
  36. H. Lang, Eisensteinsche Reihen höherer Stufe im Falle der komplexen Multiplikation, Abh. Math. Sem. Univ. Hamburg 35 (3-4), 242-250, 1971.
  37. H. Lang, Kummersche Kongruenzen für die normierten Entwicklungskoeffizienten der Weierstrassschen ℘-Funktionen, Abh. Math. Sem. Univ. Hamburg 33, 183-196, 1969.
  38. S. Lang, Elliptic functions, Addison-Wesley, 1973.
  39. C. Meyer, Über einige Anwendungen Dedekindscher Summen, J.Reine Angew. Math. 198, 143-203, 1957.
  40. C. Meyer, Über die Bildung von Klasseninvarianten binarer quadratischer Formen mittels Dedekinkscher Summen, Abh. math. Sem. Univ. Hamburg. 27 Heft 3/4, 206-230, 1964.
  41. H. RADEMACHER, Some remarks on certain generalized Dedekind sums, Acta Arith. 9, 97-105, 1964.
  42. R.Sczech, Dedekindsummen mit elliptischen Funktionen, Invent.math. 76, 523-551, 1984.
  43. C.L. Siegel, Lectures on advanced analytic number theory, Tata Bombay, 1964.
  44. A. Srivastav, M.J. Taylor, Elliptic curves with complex multiplication and Galois module structure, Invent. Math. t. 99, 165-184, 1990.
  45. H. Vandiver, General congruences involving the Bernoulli numbers, Proc. Nat. Acad. Sci. U.S.A 28, 324-328, 1942.
  46. K. G. C. von Staudt, Beweis eines Lehrsatzes die Bernoulli’schen Zahlen betreffend, J. Reine Angew. Math. 21, 372-374, 1840.
  47. F.R. Villegas, The congruences of Clausen-von Staudt and Kummer for half-integral weight Eisenstein series, Math. Nachr. 162, 187-191, 1993.
  48. H. Weber, Lehrbuch der Algebra III, Chelsea, 1908.
  49. A. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergeb. der Math. 88, Springer-Verlag, 1976.
  50. U. Weselmann, EisensteinKohomologie und Dedekindsummen für GL2 über imaginär-quadratischen Zahlenkörpern, J. Reine. Angew. Math. 389, 90-121, 1988.
  51. D. Zagier, Higher order Dedekind sums, Math.Ann. 202, 149-172, 1973.
  52. D. Zagier, A Kronecker Limit Formula for Real Quadratic Fields, Math. Ann. 213, 153-184, 1975.
  53. D. Zagier, Periods of modular forms and Jacobi theta functions, Invent.math. 104, 449-465, 1991.