Non-commutative Two Dimensional Modular Symbol

Abstract

We define iteration over a two dimensional manifold as analog of iteration over a path defined by Chen. We give several applications. Some of them include constructions of non-abelian modular symbol for $SL(3,\mathbb{Z})$ and for $S{L}_{2/K}$, where $K$ is a real quadratic field. We construct the latter as generating series of iterated integrals over a two-dimensional manifold of Hilbert modular forms. We give a motivic interpretation of this non-abelian modular symbol. Other applications give motivic interpretations of certain iterated completed Dedekind zeta functions.

27 pages