Periods of iterated integrals of holomorphic forms on a compact Riemann surface

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Abstract
Holomorphic forms are integrated iteratedly along paths in a compact Riemann surface of genus , thus inducing a homomorphism from the fundamental group to a proper multiplicative subgroup of the group of units in , where denotes the space of holomorphic forms on is the complex dual of , means the associated tensor algebra and 11'' means completion with respect to the natural grading. The associated homomorphisms from to reduces to the classical case when . We show that the images of are always cocompact in and are discrete for all if and only if the Jacobian variety of is isogenous to for some elliptic curve with complex multiplication.

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