The Hodge-de~Rham Theory of Modular Groups

By R. Hain. 2014

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This paper is an exposition of the completion of a modular group with respect to its inclusion into SL_2(Q) and the connection with the theory of modular forms and variations of mixed Hodge structure over modular curves. Among the goals of this paper are to give a context to Manin's iterated Shimura integrals (iterated integrals of modular forms) and to study relations in the "maximal Eisenstein quotient" of the completion of SL_2(Z). In particular, we use a computation of Terasoma to prove that Pollack's quadratic relations in the derivation Lie algebra of a rank two free Lie algebra lift to this Eisenstein quotient, and are thus motivic. We also construct the normal functions associated to Hecke eigen cusp forms.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e110501e74bb090da11b2a2ba8c95eb422ff815