Reciprocity of Dedekind sums and the Euler class
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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.References
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Additional Information
- Claire Burrin
- Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854
- MR Author ID: 1186967
- Email: claire.burrin@rutgers.edu
- Received by editor(s): November 29, 2016
- Published electronically: December 18, 2017
- Communicated by: Kathrin Bringmann
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1367-1376
- MSC (2010): Primary 11F20; Secondary 30F35, 20J06
- DOI: https://doi.org/10.1090/proc/13880
- MathSciNet review: 3754325