Sommes de Dedekind elliptiques et formes de Jacobi

Bayad, Abdelmejid

Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 29-42.

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 $ℂ/L$, on retrouve les résultats de S. Egami.

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 $D_L(z,\varphi )$ 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.

Classification : 11M36, 11F50, 11F20, 11A15, 11G16, 11F67, 14K25, 55N91, 55N34
Mot clés : sommes de Dedekind, formes de Jacobi, eta, loi de réciprocité, fonction thêta, fonction de Klein, fonction de Weierstrass, formule des résidus, classes de cohomologie
Keywords: Dedekind sums, Jacobi forms, eta, reciprocity law, theta function, Klein function, Weierstrass function, residues formula, cohomology classes

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