Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Elliptic Apostol sums and their reciprocity laws

Authors: Shinji Fukuhara and Noriko Yui
Journal: Trans. Amer. Math. Soc. 356 (2004), 4237-4254
MSC (2000): Primary 11F20; Secondary 33E05, 11F11
Published electronically: May 10, 2004
MathSciNet review: 2058844
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter $\tau$ having positive imaginary part. When $\tau\to i\infty$, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable $\tau$. We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).

References [Enhancements On Off] (What's this?)

  • 1. Apostol, T. M.: Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Math. J. 17, 147-157 (1950). MR 11:641g
  • 2. Apostol, T. M.: Theorems on generalized Dedekind sums. Pacific J. Math. 2, 1-9 (1952). MR 13:725c
  • 3. Bayad, A.: Sommes de Dedekind elliptiques et formes de Jacobi. Ann. Inst. Fourier, Grenoble 51, 29-42 (2001). MR 2002d:11049
  • 4. Berndt, B. C.: Reciprocity theorems for Dedekind sums and generalizations. Advances in Math. 23, 285-316 (1977). MR 55:2722
  • 5. Berndt, B. C., Evans, R. J.: On Rademacher's multiplier system for the classical theta-function. Contemp. Math. 166, 1-7 (1994) MR 96b:11060
  • 6. Carlitz, L.: On reciprocity theorem for Dedekind sums. Pacific J. Math. 3, 523-527 (1952) MR 15:12c
  • 7. Dieter, U.: Cotangent sums, a further generalization of Dedekind sums. J. Number Theory 18, 289-305 (1984). MR 85i:11042
  • 8. Egami, S.: An elliptic analogue of the multiple Dedekind sums. Compos. Math. 99, 99-103 (1995). MR 96g:11040
  • 9. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Boston-Basel-Stuttgart: Birkhäuser 1985. MR 86j:11043
  • 10. Fukuhara, S.: Modular forms, generalized Dedekind symbols and period polynomials. Math. Ann. 310, 83-101 (1998). MR 99k:11061
  • 11. Fukuhara, S.: Dedekind symbols associated with J-forms and their reciprocity law. J. Number Theory 98, 236-253 (2003).
  • 12. Fukuhara, S.: New trigonometric identities and generalized Dedekind sums. Tokyo J. Math. 26, 1-14 (2003).
  • 13. Hirzebruch, F., Berger, T., Jung, R.: Manifolds and Modular Forms. Bonn: Vieweg 1992. MR 94d:57001
  • 14. Ito, H.: On a property of elliptic Dedekind sums. J. Number Theory 27, 17-21 (1987). MR 89a:11050
  • 15. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms (Graduate Texts in Math. 97). Springer-Verlag 1993. MR 94a:11078
  • 16. Mikolás, M.: On certain sums generating the Dedekind sums and their reciprocity laws. Pacific J. Math. 7, 1167-1178 (1957). MR 19:943c
  • 17. Mordell, L. J.: Lattice points in a tetrahedron and generalized Dedekind sums. J. London Math. Soc. (N.S.) 15, 41-46 (1951). MR 13:322b
  • 18. Rademacher, H., Grosswald, E.: Dedekind Sums (Carus Math. Mono. No. 16). Math. Assoc. Amer. 1972. MR 50:9767
  • 19. Sczech, R.: Dedekindsummen mit elliptischen Funktionen. Invent. Math. 67, 523-551 (1984). MR 86h:11037
  • 20. Zagier, D.: Higher dimensional Dedekind sums. Math. Ann. 202, 149-172 (1973). MR 50:9801

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11F20, 33E05, 11F11

Retrieve articles in all journals with MSC (2000): 11F20, 33E05, 11F11

Additional Information

Shinji Fukuhara
Affiliation: Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187-8577, Japan

Noriko Yui
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6

Keywords: Generalized Dedekind sums (Apostol sums), elliptic functions, elliptic Apostol sums, modular forms, reciprocity laws
Received by editor(s): September 30, 2002
Received by editor(s) in revised form: August 7, 2003
Published electronically: May 10, 2004
Additional Notes: The first author was partially supported by Grant-in-Aid for Scientific Research (C)12640089, Ministry of Education, Sciences, Sports and Culture, Japan.
The second author was partially supported by a Research Grant from NSERC, Canada.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society