Résume | Several years ago Manin defined a non-commutative modular symbolfor subgroups of finite index of SL(2,Z), using Chen's iteratedintegrals of modular forms. I will explain its meaning. I willpresent a modification of Chen's iterated integrals, which candefine a generalization of non-commutative modular symbol onHilbert modular surface. I will explain what is a Hilbert modularsurface and what is the relation of the defined symbol to theclassical modular symbol over such a surface. As an application, Iconstruct geometric compositions of different non-commutativemodular symbols. |