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Sommes de Dedekind elliptiques et formes de Jacobi. (Elliptic Dedekind sums and Jacobi forms). (French) Zbl 1034.11030

Summary: We introduce an elliptic analogue of the multiple Dedekind sums investigated by D. Zagier [Invent. Math. 104, 449–465 (1991; Zbl 0742.11029)]. 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 a 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 [Compos. Math. 99, 99–103 (1995; Zbl 0838.11029)].

MSC:

11F20 Dedekind eta function, Dedekind sums
11F50 Jacobi forms
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
55N34 Elliptic cohomology
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