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.

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

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

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

[21] H. Ito A function on the upper halfspace which is analogous to imaginary of $\text{log}\eta \left(z\right)$, J. reine angew. Math., Volume 373 (1987), pp. 148-165 | DOI | EuDML | MR | Zbl

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

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

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

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