Dedekind sums, reciprocity, and non-arithmetic groups

TL;DR

This paper surveys recent developments in Dedekind sums, focusing on their properties and generalizations for non-uniform lattices in SL_2(R), highlighting their broad applications across mathematics and physics.

Contribution

It provides a comprehensive overview of recent advances in Dedekind sums and symbols for non-uniform lattices, connecting their properties to various mathematical fields.

Findings

Dedekind sums are connected to diverse mathematical areas.

Recent generalizations extend Dedekind sums to non-uniform lattices.

Dedekind symbols have broad applications in geometry, topology, and physics.

Abstract

Dedekind sums, arithmetic correlation sums that arose in Dedekind's study of the modular transformation of the logarithm of the eta-function, are surprisingly ubiquitous. Their arithmetic properties attracted the attention of number theorists, combinatorists, and theoretical computer scientists alike, and they appear more broadly in geometry, topology, and physics. Accordingly, there is a similarly vast literature on variations and generalizations of Dedekind sums. This note surveys recent developments of some aspects of Dedekind sums for (non-uniform) lattices in SL_2(R), otherwise referred to as Dedekind symbols.