Iterated Shimura Integrals

In this paper I continue the study of iterated integrals of modular forms and noncommutative modular symbols for Γ ⊂ SL(2,Z) started in [Ma3]. Main new results involve a description of the iterated Shimura cohomology and the image of the iterated Shimura cocycle class inside it. The concluding section of the paper contains a concise review of the classical modular symbols for SL(2) and a discussion of open problems. §0. Introduction Let M be a linearly connected space, and G a group acting on it. Then G acts on the fundamental groupoid of M thus creating a situation where the well known formalism of cohomology of G with noncommutative coefficients applies. If M is a differentiable manifold, Chen iterated integrals produce a representation of the fundamental groupoid so that we get relations between such integrals reflecting the action of G. In [Ma3] I have studied this situation for the case when M is the upper complex half plane partially completed by cusps, and the iterated integrals involve cusp forms (and eventually Eisenstein series). The questions asked and the form of answers I would like to get in this case were motivated by Drinfeld’s associators and the classical theory of ordinary integrals including the basics of Mellin transform and modular symbols. Here I continue this study, stressing the Shimura approach to the SL(2)–modular symbols of arbitrary weight and attempting its iterated extension. The paper is structured as follows. In §1 the notation and some background of noncommutative group cohomology is reviewed. In §2 the theory of the iterated Shimura cocycle is given. Finally, §3 sketches the classical theory of modular symbols and discusses open problems. The reader might prefer to read this section first, as a motivation for our attempt to produce its iterated version.