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Reciprocity of Dedekind sums and the Euler class

Author: Claire Burrin
Journal: Proc. Amer. Math. Soc. 146 (2018), 1367-1376
MSC (2010): Primary 11F20; Secondary 30F35, 20J06
Published electronically: December 18, 2017
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Abstract: Dedekind sums are arithmetic sums that were first introduced by Dedekind in the context of elliptic functions and modular forms, and later recognized to be surprisingly ubiquitous. Among the variations and generalizations introduced since, there is a construction of Dedekind sums for lattices in $ \mathrm {SL}_2(\mathrm {R})$. Building upon work of Asai, we prove the reciprocity law for these Dedekind sums, based on a concrete realization of the Euler class. As an application, we obtain an explicit formula for Dedekind sums on Hecke triangle groups in terms of continued fractions.

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Additional Information

Claire Burrin
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854

Received by editor(s): November 29, 2016
Published electronically: December 18, 2017
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2017 American Mathematical Society

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