Multiple Dedekind Symbols

TL;DR

This paper explores the theory of multiple Dedekind symbols and reciprocity functions, establishing a bijection, defining their products, and connecting them to modular forms via iterated integrals, with a focus on their shuffle properties.

Contribution

It introduces a bijection between multiple Dedekind symbols and reciprocity functions, and constructs shuffled versions from modular forms using regularized iterated integrals.

Findings

Established a bijection between Dedekind symbols and reciprocity functions

Defined products of reciprocity functions and analyzed their shuffle properties

Constructed and computed shuffled symbols from modular forms

Abstract

In this paper we study multiple Dedekind symbols and the associated multiple reciprocity functions. There is a bijection between the two sets of them after a normalization. By this bijection we define products of multiple reciprocity functions, and study the relationship to the shuffle property. We construct and calculate shuffled multiple Dedekind symbols and shuffled multiple reciprocity functions from modular forms by regularized iterated integrals. Also we give a decomposition for them.