Non-commutative Hilbert modular symbols

TL;DR

This paper introduces non-commutative Hilbert modular symbols, extending classical and non-commutative modular symbols, and explores their relation to periods, multiple zeta values, and higher-dimensional iterated integrals.

Contribution

It constructs non-commutative Hilbert modular symbols using higher-dimensional iterated integrals called membranes, generalizing Manin's approach and linking to multiple Dedekind zeta values.

Findings

Non-commutative Hilbert modular symbols are periods in the sense of Kontsevich-Zagier.

Hecke operators act naturally on these symbols.

Formulas relate non-commutative Hilbert modular symbols to multiple Dedekind zeta values.

Abstract

The main goal of this paper is to construct non-commutative Hilbert modular symbols. However, we also construct commutative Hilbert modular symbols. Both the commutative and the non-commutative Hilbert modular symbols are generalizations of Manin's classical and non-commutative modular symbols. We prove that many cases of (non-)commutative Hilbert modular symbols are periods in the sense on Kontsevich-Zagier. Hecke operators act naturally on them. Manin defines the non-commutative modilar symbol in terms of iterated path integrals. In order to define non-commutative Hilbert modular symbols, we use a generalization of iterated path integrals to higher dimensions, which we call iterated integrals on membranes. Manin examines similarities between non-commutative modular symbol and multiple zeta values both in terms of infinite series and in terms of iterated path integrals. Here we examine similarities in the formulas for non-commutative Hilbert modular symbol and multiple Dedekind zeta values, recently defined by the author, both in terms of infinite series and in terms of iterated integrals on membranes.