Noncommutative Hilbert modular symbols

Horozov, Ivan

doi:10.2140/ant.2015.9.317

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Noncommutative Hilbert modular symbols

Ivan Horozov

Vol. 9 (2015), No. 2, 317–370

DOI: 10.2140/ant.2015.9.317

Abstract

The main goal of this paper is to construct noncommutative Hilbert modular symbols. However, we also construct commutative Hilbert modular symbols. Both the commutative and the noncommutative Hilbert modular symbols are generalizations of Manin’s classical and noncommutative modular symbols. We prove that many cases of (non)commutative Hilbert modular symbols are periods in the Kontsevich–Zagier sense. Hecke operators act naturally on them.

Manin defined the noncommutative modular symbol in terms of iterated path integrals. In order to define noncommutative Hilbert modular symbols, we use a generalization of iterated path integrals to higher dimensions, which we call iterated integrals on membranes. Manin examined similarities between noncommutative modular symbol and multiple zeta values in terms of both infinite series and of iterated path integrals. Here we examine similarities in the formulas for noncommutative Hilbert modular symbol and multiple Dedekind zeta values, recently defined by the current author, in terms of both infinite series and iterated integrals on membranes.

Keywords

modular symbols, Hilbert modular groups, Hilbert modular surfaces, iterated integrals

Mathematical Subject Classification 2010

Primary: 11F41

Secondary: 11F67, 11M32

Milestones

Received: 22 August 2013

Revised: 17 September 2014

Accepted: 26 November 2014

Published: 5 March 2015

Authors

Ivan Horozov
Department of Mathematics Washington University in St. Louis One Brookings Drive Campus Box 1146 Saint Louis, MO 63130 United States

Vol. 9, No. 2, 2015