Sommes de Dedekind elliptiques et formes de Jacobi
Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 29-42.

À partir des formes de Jacobi D L (z,ϕ), 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 ϕ du tore complexe /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,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.

DOI : https://doi.org/10.5802/aif.1813
Classification : 11M36,  11F50,  11F20,  11A15,  11G16,  11F67,  14K25,  55N91,  55N34
Mots 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
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     author = {Bayad, Abdelmejid},
     title = {Sommes de Dedekind elliptiques et formes de Jacobi},
     journal = {Annales de l'Institut Fourier},
     pages = {29--42},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {1},
     year = {2001},
     doi = {10.5802/aif.1813},
     zbl = {1034.11030},
     mrnumber = {1821066},
     language = {fr},
     url = {www.numdam.org/item/AIF_2001__51_1_29_0/}
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Bayad, Abdelmejid. Sommes de Dedekind elliptiques et formes de Jacobi. Annales de l'Institut Fourier, Tome 51 (2001) no. 1, pp. 29-42. doi : 10.5802/aif.1813. http://www.numdam.org/item/AIF_2001__51_1_29_0/

[1] T. M. Apostol Introduction to analytic Number Theory, Springer-Verlag, New York, 1976 | MR 434929 | Zbl 0335.10001

[2] M.F. Atiyah; F. Hirzebruch Cohomologie-Operationen und charakteristische Klassen, Math. Z., Volume 77 (1961), pp. 149-187 | Article | MR 156361 | Zbl 0109.16002

[3] M.F. Atiyah; F. Hirzebruch Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc., Volume 65 (1959), pp. 276-281 | Article | MR 110106 | Zbl 0142.40901

[4] M.F. Atiyah; I.M. Singer The index of elliptic Operators, Ann. of. Math., Volume 87 (1968), pp. 546-604 | Article | MR 236952 | Zbl 0164.24301

[5] A. Bayad; G. Robert Amélioration d'une congruence pour certains éléments de Stickelberger quadratiques, Bull. Soc. Math. France, Volume 125 (1997), pp. 249-267 | Numdam | MR 1478032 | Zbl 0895.11021

[6] A. Bayad; G. Robert Note sur une forme de Jacobi méromorphe, C.R.A.S., Paris, Volume 325 (1997), pp. 455-460 | MR 1692306 | Zbl 0885.11035

[7] H. Cohen Sommes de carrés, fonctions L et formes modulaires, C.R.A.S., Paris, Volume 277 (1973), pp. 827-830 | MR 374061 | Zbl 0267.10066

[8] H. Cohen Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann., Volume 217 (1975), pp. 271-285 | Article | MR 382192 | Zbl 0311.10030

[9] R. Dedekind Erlauterungen zu zwei Fragmenten von Riemann, Ges. math. Werke, Volume erster Band (1930), pp. 159-173

[10] U. Dieter Pseudo-random numbers: The exact distribution of pairs, Math. of Computation, Volume 25 (1971) | MR 298727 | Zbl 0257.65010

[11] S. Egami An elliptic analogue of multiple Dedekind sums, Compositio Math., Volume 99 (1995), pp. 99-103 | EuDML 90409 | Numdam | MR 1352569 | Zbl 0838.11029

[12] M. Eichler; D. Zagier The Theory of Jacobi forms, Progress in Math., Volume 55, Birkhäuser, 1985 | MR 781735 | Zbl 0554.10018

[13] E. Grosswald; H. Rademacher Dedekind Sums, Carus Mathematical Monographs, Volume No. 16, Mathematical Assoc. America, Washington D.C, 1972 | MR 357299 | Zbl 0251.10020

[14] G. Harder Periods Integrals of Cohomology Classes which are represented by Eisenstein Series, Proc. Bombay Colloquium (1979) | MR 633658 | Zbl 0497.22021

[15] G. Harder; Koblitz, N (ed.) Periods Integrals of Eisenstein Cohomology Classes which and special values of somes L-functions, Number theory related to Fermat's last theorem (1982), pp. 103-142 | MR 685293 | Zbl 0517.12008

[16] G. H. Hardy; S. Ramanujan Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2), Volume 17 (1918), pp. 75-115 | Article | JFM 46.0198.04 | MR 1575586

[17] F. Hirzebruch Topological methods in algebraic geometry, Springer, Berlin--Heidelberg--New York, 1966 | MR 202713 | Zbl 0376.14001

[18] F. Hirzebruch The signature theorem: reminiscences and recreation, Prospects in Mathematics (Ann. of Math. Studies) Volume 70 (1971), pp. 3-31 | MR 368023 | Zbl 0252.58009

[19] F. Hirzebruch; T. Berger; R. Jung Manifolds and Modular forms, Aspects of Math., Volume E20, Vieweg, 1992 | MR 1189136 | Zbl 0767.57014

[20] F. Hirzebruch; D. Zagier The Atiyah-Singer Theorem and Elementary Number Theory, Math. Lecture Series, Volume 3, Publish or Perish Inc., 1974 | MR 650832 | Zbl 0288.10001

[21] H. Ito A function on the upper halfspace which is analogous to imaginary of logη(z), J. reine angew. Math., Volume 373 (1987), pp. 148-165 | Article | EuDML 152888 | MR 870309 | Zbl 0601.10021

[22] H. Ito On a property of elliptic Dedekind sums, J. Number Th., Volume 27 (1987), pp. 17-21 | Article | MR 904003 | Zbl 0624.10018

[23] D. Kubert Product formulae on elliptic curves, Inv. Math., Volume 117 (1994), pp. 227-273 | Article | EuDML 144215 | MR 1273265 | Zbl 0834.14016

[24] D. Kubert; S. Lang Modular units, Grundlehren der Math. Wiss., Volume 244, Springer-Verlag, 1981 | MR 648603 | Zbl 0492.12002

[25] P. S. Landweber Elliptic Curves and Modular Forms in Algebraic Topology, Proceeding Princeton 1986 (Lectures Notes in Mathematics) Volume 1362 (1988) | Zbl 0642.00007

[26] S. Lang Elliptic functions, Addison-Wesley, 1973 | MR 409362 | Zbl 0316.14001

[27] C. Meyer Uber einige Anwendungen Dedekindscher Summen, J. reine angew. Math., Volume 198 (1957), pp. 143-203 | Article | EuDML 150351 | MR 104643 | Zbl 0079.10303

[28] C. Meyer Uber die Bildung von Klasseninvarianten binärer quadratischer Formen mittels Dedekinkscher Summen, Abh. Math. Sem. Univ. Hamburg, Volume 27 (1964) no. Heft 3/4, pp. 206-230 | Article | MR 170865 | Zbl 0122.05201

[29] L.J. Mordell The reciprocity formula for Dedekind sums, Amer. J. Math., Volume 73 (1951), pp. 593-598 | Article | MR 42449 | Zbl 0042.27401

[30] D. Mumford Tata Lectures on Theta I, Progress in Math., Volume 28, Birkhäuser, 1983 | MR 688651 | Zbl 0509.14049

[31] H. Rademacher On the partition function p(n), Proc. London Math. Soc. (2), Volume 43 (1937), pp. 241-254 | Article | JFM 63.0140.02

[32] R. Sczech Dedekindsummen mit elliptischen Funktionen, Invent. Math., Volume 76 (1984), pp. 523-551 | Article | EuDML 143135 | MR 746541 | Zbl 0521.10021

[33] D. Zagier Periods of modular forms and Jacobi theta functions, Invent. Math., Volume 104 (1991), pp. 449-465 | Article | EuDML 143891 | MR 1106744 | Zbl 0742.11029

[34] D. Zagier Higher order Dedekind sums, Math. Ann., Volume 202 (1973), pp. 149-172 | EuDML 162407 | MR 357333 | Zbl 0237.10025

[35] D. Zagier Note on the Landweber-Stong Elliptic Genus (Lectures Notes in Mathematics) Volume 1362 (1988), pp. 216-224 | Zbl 0653.57016