Algebra & Number Theory

Noncommutative Hilbert modular symbols

Ivan Horozov

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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.

Article information

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

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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

modular symbols Hilbert modular groups Hilbert modular surfaces iterated integrals


Horozov, Ivan. Noncommutative Hilbert modular symbols. Algebra Number Theory 9 (2015), no. 2, 317--370. doi:10.2140/ant.2015.9.317.

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