Algebra & Number Theory
- Algebra Number Theory
- Volume 9, Number 2 (2015), 317-370.
Noncommutative Hilbert modular symbols
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.
Algebra Number Theory, Volume 9, Number 2 (2015), 317-370.
Received: 22 August 2013
Revised: 17 September 2014
Accepted: 26 November 2014
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Horozov, Ivan. Noncommutative Hilbert modular symbols. Algebra Number Theory 9 (2015), no. 2, 317--370. doi:10.2140/ant.2015.9.317. https://projecteuclid.org/euclid.ant/1510842284