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March 25, 2015 By

Rationality of Dedekind Sums in Finite Fields

C. R. Math. Rep. Acad. Sci. Canada Vol. 34 (4) 2012, pp. 105–111

December 31, 2012

Yoshinori Hamahata, Institute for Teaching and Learning, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan; e-mail: hamahata@fc.ritsumei.ac.jp

Abstract/Résumé:

In Higher dimensional Dedekind sums in finite fields. Finite Fields Appl. 18 (2012), 19–25, we introduced the Dedekind–Zagier sum in finite fields. It is defined by a lattice $\Lambda$. The objective of this paper is to present a criterion for the rationality of our Dedekind–Zagier sum. For this purpose, we establish a connection between the field of definition of the exponential function for $\Lambda$ and the field of definition of the Dedekind–Zagier sum for $\Lambda$.

Dans Higher dimensional Dedekind sums in finite fields. Finite Fields Appl. 18 (2012), 19–25, nous avons introduit la somme de Dedekind–Zagier dans des corps finis. La somme est définie à partir d’un réseau $\Lambda$. L’objectif de ce travail est de présenter un critère de la rationalité de notre somme de Dedekind–Zagier. Pour le but, nous éstablissons la connexion entre le corps de définition de la fonction exponentielle pour $\Lambda$ et le corps de définition de notre somme de Dedekind–Zagier pour $\Lambda$.

Keywords: Dedekind sums, finite fields, lattices
AMS Subject Classification: Dedekind eta function; Dedekind sums 11F20

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