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Multiple modular values are a common generalisation of multiple zeta valuesand periods of modular forms, and are periods of a hypothetical Tannakiancategory of mixed modular motives. They are given by regularised iteratedintegrals on the upper half plane generalising the iterated Shimura integralsof Manin. In this paper, some first properties of the underlying theory areestablished in the case of the full modular group: in particular, therelationship with special values of L-functions of modular forms at allpositive integers; and the action of the conjectural motivic Galois group via acertain group of automorphisms.

Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$