Abstract
Okada (J Number Theory, 130:1750–1762, 2010) introduced Dedekind sums associated to a certain A-lattice, and established the reciprocity law. In this paper, we introduce Dedekind sums for arbitrary A-lattice and establish the reciprocity law for them. We next introduce higher dimensional Dedekind sums for any A-lattice. These Dedekind sums are analogues of Zagier’s higher dimensional Dedekind sums. We discuss the reciprocity law, rationality and characterization of these sums.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \({\mathbb{F}_q}\) :
-
The finite field with q elements
- \({\mathcal{C}}\) :
-
A smooth, projective, geometrically connected curve over \({\mathbb{F}_q}\)
- \({K=\mathbb{F}_q(\mathcal{C})}\) :
-
The function field of \({\mathcal{C}}\) over \({\mathbb{F}_q}\)
- \({\infty\in\mathcal{C}}\) :
-
A closed point of degree d ∞ over \({\mathbb{F}_q}\) , or equivalently a place of K of degree d ∞
- υ ∞ :
-
The valuation associated to ∞
- \({|\, |_{\infty}}\) :
-
The normalized absolute value corresponding to υ ∞, i.e., for \({x\in K, |x|_{\infty}=q^{\deg (x)}=q^{-d_{\infty}v_{\infty}(x)}}\)
- K ∞ :
-
The completion of K with respect to \({|\, |_{\infty}}\)
- \({\overline{K_{\infty}}}\) :
-
A fixed algebraic closure of K ∞
- C ∞ :
-
The completion of \({\overline{K_{\infty}}}\)
- \({A=H^0(\mathcal{C}-\infty,\mathcal{O}_{\mathcal{C}})}\) :
-
The ring of functions regular outside ∞
- \({{\sum}'}\) :
-
The sum over non-zero elements
- \({{\prod}'}\) :
-
The product over non-zero elements
References
Deligne P., Husemöller D.: Survey of Drinfeld modules. Contemp. Math. 67, 25–91 (1987)
Gekeler E.-U.: On the coefficients of Drinfeld modular forms. Invent. Math. 93, 667–700 (1988)
Goss D.: The algebraist’s upper half-planes. Bull. Am. Math. Soc. 2, 391–415 (1980)
Goss D.: Basic Structures of Function Fields. Springer, Berlin (1996)
Hamahata, Y.: Dedekind sums for finite fields. In: Diophantine Analysis and Related Fields: DARF 2007/2008. AIP conference proceedings, vol. 976, pp. 96–102. American Institute of Physics (2008)
Hamahata, Y.: Finite Dedekind sums. In: Arithmetic of Finite Fields. Lecture Notes in Computer Science, vol. 5130, pp. 11–18. Springer, Berlin (2008)
Hamahata Y.: Dedekind sums in finite characteristic. Proc. Japan Acad. Ser. A 84, 89–92 (2008)
Okada S.: Analogies of Dedekind sums in function fields. J. Number Theory 130, 1750–1762 (2010)
Rademacher H., Grosswald E.: Dedekind Sums. The Mathematical Association of America, Washington (1972)
Sczech R.: Dedekindsummen mit elliptischen Funktionen. Invent. Math. 76, 523–551 (1984)
Zagier D.: Higher-dimensional Dedekind sums. Math. Ann. 202, 149–172 (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by U. Zannier.
Dedicated to Professor Tomoyoshi Ibukiyama.
Rights and permissions
About this article
Cite this article
Hamahata, Y. Dedekind sums in function fields. Monatsh Math 167, 461–480 (2012). https://doi.org/10.1007/s00605-012-0423-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-012-0423-8