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Abstract |
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Modular symbols are integrals of modular forms over a region connecting cusp points. A few years ago Manin has defined non-commutative modular symbols for finite index subgroups of SL(2,Z). His method is based on iterated integrals over a path as defined by Chen. Manin iterates modular forms and the generating series of all such integrals, which he calls non-commutative modular symbol, has interesting properties. My construction consist of two parts. First, I define iterated integrals over a (real) two-dimensional region so, that all such iterated integrals naturally form a Hopf algebra. Then I apply this construction to Hilbert modular surfaces. Using this new type of iteration, I construct non-commutative two-dimensional modular symbols, which are group-like elements in the Hopf algebra. |