Higher dimensional Dedekind sums in finite fields
The \textit{Dedekind sum} can be defined by \[ s(h,k) := {1 \over {4k}} \sum_{ r=1 }^{ k-1 } \cot \left( {{\pi r} \over k} \right) \cot \left( {{\pi hr} \over k} \right) , \] where \(h\) and \(k\) are positive, relatively prime integers. Dedekind sums first appeared in Dedekind's work on the \(\eta\)-function and have been generalized in various ways, e.g., by \textit{D. Zagier} [Math. Ann. 202, 149--172 (1973; Zbl 0237.10025)] who considered an arbitrary number of cotangent factors. The paper under review continues the authors' study of Dedekind sums in finite fields [\textit{Y. Hamahata}, AIP Conf. Proc. 976, 96--102 (2008; Zbl 1171.11025)]; their analogue of the classical Dedekind sum is \[ s_\Lambda (h,k) := {1 \over k} \sum_{ \lambda \in \Lambda } \!{}^\prime \;e_\Lambda \left( {{ h \lambda } \over k} \right)^{ -1 } e_\Lambda \left( {{ \lambda } \over k} \right)^{-1} \] where \(\Lambda\) is a lattice in \({\mathbb F}_q\) and \[ e_\Lambda (z) := z \prod_{ \lambda \in \Lambda } \!{}^\prime \left( 1 - {z \over \lambda} \right). \] Here the \('\) indicates that the sum/product is only taken over nonzero and nonsingular values. (Note that \(e_\Lambda (z)^{ -1 } \) is a natural finite-field analogue of the cotangent function.) The authors introduce a version of their Dedekind sum in finite fields that has an arbitrary number of factors, analogous to Zagier's higher-dimensional Dedekind sum mentioned above. The main result is a \textit{reciprocity theorem}: under certain (mild) conditions, the sum of higher-dimensional finite-field Dedekind sums, whose arguments are cyclically permuted, equals a simple expression involving Eisenstein series for~\(\Lambda\).
scientific article; zbMATH DE number 6005612
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