Non-commutative Hilbert modular symbols

Ivan Horozov

doi:10.2140/ant.2015.9.317

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arXiv · 1308.4991

Non-commutative Hilbert modular symbols

Ivan Horozov

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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.

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Ivan Horozov. 2014-09-23. Non-commutative Hilbert modular symbols. https://doi.org/10.2140/ant.2015.9.317

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