no. 2
A generalization of the reciprocity law of multiple Dedekind sums
[Une généralisation de la loi de reciprocité des sommes multiples de Dedekind]
Asano, Masahiro1
1 Nagoya University Graduate School of Mathemactics Chikusa-ku, Nagoya 464-8602 (Japan)
Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 361-377.

Plusieurs sommes multiples de Dedekind ont été introduites par B.C.Berndt, L.Carlitz, S.Egami, D.Zagier et A.Bayad. Dans cet article, après avoir remarqué la forme de Jacobi dans Bayad [4], la fonction cotangente dans Zagier [23], le résultat d’Egami sur les fonctions cotangentes [14] et leurs lois de reciprocité, nous étudions un cas spécial de la forme de Jacobi de Bayad [4] et déduisons une généralisation du résultat d’Egami sur les fonctions cotangentes et une généralisation du résultat de Zagier. De plus, nous considérons leurs lois de réciprocité.

Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad.

In this paper, noticing the Jacobi form in Bayad [4], the cotangent function in Zagier [23], Egami’s result on cotangent functions [14] and their reciprocity laws, we study a special case of the Jacobi forms in Bayad [4] and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws.

DOI : 10.5802/aif.2261
Classification : 11A15, 11B68, 11F20, 11F23, 11F50
Keywords: Dedekind sums, reciprocity law, Jacobi forms
Mot clés : somme de Dedekind, loi de reciprocité, formes de Jacobi

Asano, Masahiro 1

1 Nagoya University Graduate School of Mathemactics Chikusa-ku, Nagoya 464-8602 (Japan) 
@article{AIF_2007__57_2_361_0,
     author = {Asano, Masahiro},
     title = {A generalization of the reciprocity law of multiple {Dedekind} sums},
     journal = {Annales de l'Institut Fourier},
     pages = {361--377},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     doi = {10.5802/aif.2261},
     mrnumber = {2310944},
     zbl = {1158.11022},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2261/}
}
TY  - JOUR
AU  - Asano, Masahiro
TI  - A generalization of the reciprocity law of multiple Dedekind sums
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 361
EP  - 377
VL  - 57
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2261/
DO  - 10.5802/aif.2261
LA  - en
ID  - AIF_2007__57_2_361_0
ER  - 
%0 Journal Article
%A Asano, Masahiro
%T A generalization of the reciprocity law of multiple Dedekind sums
%J Annales de l'Institut Fourier
%D 2007
%P 361-377
%V 57
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2261/
%R 10.5802/aif.2261
%G en
%F AIF_2007__57_2_361_0
Asano, Masahiro. A generalization of the reciprocity law of multiple Dedekind sums. Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 361-377. doi : 10.5802/aif.2261. https://aif.centre-mersenne.org/articles/10.5802/aif.2261/

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

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

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

[4] Bayad, A. Sommes de Dedekind elliptiques et formes de Jacobi, Ann. Inst. Fourier, Volume 51 (2001) no. 1, pp. 29-42 | DOI | Numdam | MR | Zbl

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

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

[7] Beck, M. Dedekind cotangent sums, Acta Arith., Volume 109 (2003) no. 2, pp. 109-130 | DOI | MR | Zbl

[8] Berndt, B. C. Reciprocity theorems for Dedekind sums and generalizations, Advances in Math., Volume 23 (1977), pp. 285-316 | DOI | MR | Zbl

[9] Berndt, B. C.; Dieter, U. Sums involving the greatest integer function and Riemann-Stieltjes integration, J. Reine Angew. Math., Volume 337 (1982), pp. 208-220 | MR | Zbl

[10] Carlitz, L. A note on generalized Dedekind sums, Duke Math. J., Volume 21 (1954), pp. 399-404 | DOI | MR | Zbl

[11] Carlitz, L. A theorem on generalized Dedekind sums, Acta Arith., Volume 11 (1965), pp. 253-260 | MR | Zbl

[12] Carlitz, L. Many term relations for multiple Dedekind sums, Indian J. Math., Volume 20 (1978), pp. 77-89 | MR | Zbl

[13] Dieter, U. Pseudo-random numbers : the exact distribution of pairs, Math. of Computation, Volume 25 (1971), pp. 855-883 | MR | Zbl

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

[15] Fukuhara, S.; Yui, N. Elliptic Apostol sums and their reciprocity laws, Trans. Amer. Math. Soc., Volume 356 (2004) no. 10, pp. 4237-4254 | DOI | MR | Zbl

[16] Harder, G. Periods integrals of cohomology classes which are represented by Eisenstein series, Proc. Bombay Colloquium 1979 (1981), pp. 41-115 | MR | Zbl

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

[18] Hirzebruch, F.; Berger, T.; Jung, R. Manifolds and modular forms, Aspects of Mathematics, E20, Vieweg Verlag, 1992 | MR | Zbl

[19] Ito, H. A function on the upper half space which is analogous to imaginary part of logη(z), J. Reine Angew. Math., Volume 373 (1987), pp. 148-165 | DOI | MR | Zbl

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

[21] Rademacher, H. Generalization of the reciprocity formula for Dedekind sums, Duke Math. J., Volume 21 (1954), pp. 391-397 | DOI | MR | Zbl

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

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

Cité par Sources :