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.

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