Multiple Dedekind sums and Jacobi forms

Abdelmejid Bayad[1]

  • [1] Université d'Evry, Département de Mathématiques, boulevard des Coquibus, 91025 Evry Cedex (France)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 1, page 29-42
  • ISSN: 0373-0956

Abstract

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In this paper we introduce an elliptic analogue of the multiple Dedekind sums investigated by D. Zagier. Our method and results are quite similar to D. Zagier except the use of Jacobi forms D L ( z , ϕ ) in place of the cotangent function which appeared there. In fact we show the reciprocity law for our Dedekind sums. By limiting procedure we can recover the corresponding results on multiple Dedekind (cotangent) sums. By a specialization to the 2-division points, we can recover the known results of S. Egami.

How to cite

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Bayad, Abdelmejid. "Sommes de Dedekind elliptiques et formes de Jacobi." Annales de l’institut Fourier 51.1 (2001): 29-42. <http://eudml.org/doc/115912>.

@article{Bayad2001,
abstract = {À partir des formes de Jacobi $D_L(z,\varphi )$, on construit une somme de Dedekind elliptique. On obtient ainsi un analogue elliptique aux sommes multiples de Dedekind construites à partir des fonctions cotangentes, étudiées par D. Zagier. En outre, on établit une loi de réciprocité satisfaite par ces nouvelles sommes. Par une procédure de limite, on peut retrouver la loi de réciprocité remplie par les sommes multiples de Dedekind classiques. D’autre part, en les spécialisant en des paramètres de points de 2- division, en la seconde variable $\varphi $ du tore complexe $\{\mathbb \{C\}\}/L$, on retrouve les résultats de S. Egami.},
affiliation = {Université d'Evry, Département de Mathématiques, boulevard des Coquibus, 91025 Evry Cedex (France)},
author = {Bayad, Abdelmejid},
journal = {Annales de l’institut Fourier},
keywords = {Dedekind sums; Jacobi forms; eta; reciprocity law; theta function; Klein function; Weierstrass function; residues formula; cohomology classes},
language = {fre},
number = {1},
pages = {29-42},
publisher = {Association des Annales de l'Institut Fourier},
title = {Sommes de Dedekind elliptiques et formes de Jacobi},
url = {http://eudml.org/doc/115912},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Bayad, Abdelmejid
TI - Sommes de Dedekind elliptiques et formes de Jacobi
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 1
SP - 29
EP - 42
AB - À partir des formes de Jacobi $D_L(z,\varphi )$, on construit une somme de Dedekind elliptique. On obtient ainsi un analogue elliptique aux sommes multiples de Dedekind construites à partir des fonctions cotangentes, étudiées par D. Zagier. En outre, on établit une loi de réciprocité satisfaite par ces nouvelles sommes. Par une procédure de limite, on peut retrouver la loi de réciprocité remplie par les sommes multiples de Dedekind classiques. D’autre part, en les spécialisant en des paramètres de points de 2- division, en la seconde variable $\varphi $ du tore complexe ${\mathbb {C}}/L$, on retrouve les résultats de S. Egami.
LA - fre
KW - Dedekind sums; Jacobi forms; eta; reciprocity law; theta function; Klein function; Weierstrass function; residues formula; cohomology classes
UR - http://eudml.org/doc/115912
ER -

References

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