Reciprocity of Dedekind sums and the Euler class

TL;DR

This paper proves a reciprocity law for generalized Dedekind sums associated with lattices in SL(2,R), using the Euler class, and provides explicit formulas for Dedekind sums in Hecke triangle groups.

Contribution

It establishes the reciprocity law for Dedekind sums for lattices in SL(2,R) based on the Euler class, extending previous work and providing explicit formulas.

Findings

Proved reciprocity law for Dedekind sums in lattices of SL(2,R).

Derived explicit formulas for Dedekind sums in Hecke triangle groups.

Connected Dedekind sums with continued fractions.

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(\mathbf{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 expressing Dedekind sums for Hecke triangle groups in terms of continued fractions.