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.
