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 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.
À partir des formes de Jacobi , 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 du tore complexe , on retrouve les résultats de S. Egami.
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
@article{AIF_2001__51_1_29_0, 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{\textquoteright}institut Fourier}, volume = {51}, number = {1}, year = {2001}, doi = {10.5802/aif.1813}, mrnumber = {1821066}, zbl = {1034.11030}, language = {fr}, url = {http://www.numdam.org/articles/10.5802/aif.1813/} }
TY - JOUR AU - Bayad, Abdelmejid TI - Sommes de Dedekind elliptiques et formes de Jacobi JO - Annales de l'Institut Fourier PY - 2001 SP - 29 EP - 42 VL - 51 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1813/ DO - 10.5802/aif.1813 LA - fr ID - AIF_2001__51_1_29_0 ER -
%0 Journal Article %A Bayad, Abdelmejid %T Sommes de Dedekind elliptiques et formes de Jacobi %J Annales de l'Institut Fourier %D 2001 %P 29-42 %V 51 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1813/ %R 10.5802/aif.1813 %G fr %F AIF_2001__51_1_29_0
Bayad, Abdelmejid. Sommes de Dedekind elliptiques et formes de Jacobi. Annales de l'Institut Fourier, Volume 51 (2001) no. 1, pp. 29-42. doi : 10.5802/aif.1813. http://www.numdam.org/articles/10.5802/aif.1813/
[1] Introduction to analytic Number Theory, Springer-Verlag, New York, 1976 | MR | Zbl
[2] Cohomologie-Operationen und charakteristische Klassen, Math. Z., Volume 77 (1961), pp. 149-187 | DOI | MR | Zbl
[3] Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc., Volume 65 (1959), pp. 276-281 | DOI | MR | Zbl
[4] The index of elliptic Operators, Ann. of. Math., Volume 87 (1968), pp. 546-604 | DOI | MR | Zbl
[5] Amélioration d'une congruence pour certains éléments de Stickelberger quadratiques, Bull. Soc. Math. France, Volume 125 (1997), pp. 249-267 | Numdam | MR | Zbl
[6] Note sur une forme de Jacobi méromorphe, C.R.A.S., Paris, Volume 325 (1997), pp. 455-460 | MR | Zbl
[7] Sommes de carrés, fonctions L et formes modulaires, C.R.A.S., Paris, Volume 277 (1973), pp. 827-830 | MR | Zbl
[8] Sums involving the values at negative integers of -functions of quadratic characters, Math. Ann., Volume 217 (1975), pp. 271-285 | DOI | MR | Zbl
[9] Erlauterungen zu zwei Fragmenten von Riemann, Ges. math. Werke, Volume erster Band (1930), pp. 159-173
[10] Pseudo-random numbers: The exact distribution of pairs, Math. of Computation, Volume 25 (1971) | MR | Zbl
[11] An elliptic analogue of multiple Dedekind sums, Compositio Math., Volume 99 (1995), pp. 99-103 | EuDML | Numdam | MR | Zbl
[12] The Theory of Jacobi forms, Progress in Math., 55, Birkhäuser, 1985 | MR | Zbl
[13] Dedekind Sums, Carus Mathematical Monographs, No. 16, Mathematical Assoc. America, Washington D.C, 1972 | MR | Zbl
[14] Periods Integrals of Cohomology Classes which are represented by Eisenstein Series, Proc. Bombay Colloquium (1979) | MR | Zbl
[15] 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 | Zbl
[16] Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2), Volume 17 (1918), pp. 75-115 | DOI | JFM | MR
[17] Topological methods in algebraic geometry, Springer, Berlin--Heidelberg--New York, 1966 | MR | Zbl
[18] The signature theorem: reminiscences and recreation, Prospects in Mathematics (Ann. of Math. Studies), Volume 70 (1971), pp. 3-31 | MR | Zbl
[19] Manifolds and Modular forms, Aspects of Math., E20, Vieweg, 1992 | MR | Zbl
[20] The Atiyah-Singer Theorem and Elementary Number Theory, Math. Lecture Series, 3, Publish or Perish Inc., 1974 | MR | Zbl
[21] A function on the upper halfspace which is analogous to imaginary of , J. reine angew. Math., Volume 373 (1987), pp. 148-165 | DOI | EuDML | MR | Zbl
[22] On a property of elliptic Dedekind sums, J. Number Th., Volume 27 (1987), pp. 17-21 | DOI | MR | Zbl
[23] Product formulae on elliptic curves, Inv. Math., Volume 117 (1994), pp. 227-273 | DOI | EuDML | MR | Zbl
[24] Modular units, Grundlehren der Math. Wiss., 244, Springer-Verlag, 1981 | MR | Zbl
[25] Elliptic Curves and Modular Forms in Algebraic Topology, Proceeding Princeton 1986 (Lectures Notes in Mathematics), Volume 1362 (1988) | Zbl
[26] Elliptic functions, Addison-Wesley, 1973 | MR | Zbl
[27] Uber einige Anwendungen Dedekindscher Summen, J. reine angew. Math., Volume 198 (1957), pp. 143-203 | DOI | EuDML | MR | Zbl
[28] 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 | DOI | MR | Zbl
[29] The reciprocity formula for Dedekind sums, Amer. J. Math., Volume 73 (1951), pp. 593-598 | DOI | MR | Zbl
[30] Tata Lectures on Theta I, Progress in Math., 28, Birkhäuser, 1983 | MR | Zbl
[31] On the partition function , Proc. London Math. Soc. (2), Volume 43 (1937), pp. 241-254 | DOI | JFM
[32] Dedekindsummen mit elliptischen Funktionen, Invent. Math., Volume 76 (1984), pp. 523-551 | DOI | EuDML | MR | Zbl
[33] Periods of modular forms and Jacobi theta functions, Invent. Math., Volume 104 (1991), pp. 449-465 | DOI | EuDML | MR | Zbl
[34] Higher order Dedekind sums, Math. Ann., Volume 202 (1973), pp. 149-172 | EuDML | MR | Zbl
[35] Note on the Landweber-Stong Elliptic Genus (Lectures Notes in Mathematics), Volume 1362 (1988), pp. 216-224 | Zbl
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