Triple Products of Eisenstein Series

Abstract

In this thesis, we construct a Massey triple product on the Deligne cohomology of the modular curve with coefficients in symmetric powers of the standard representation of the modular group. This result is obtained by constructing a Massey triple product on the extension groups in the category of admissible variations of mixed Hodge structure over the modular curve, which induces the desired construction on Deligne cohomology. The result extends Brown's construction of the cup product on Deligne cohomology to a higher cohomological product.Massey triple products on Deligne cohomology have been previously investigated by Deninger, who considered Deligne cohomology with trivial real coefficients. By working over the reals, Deninger was able to compute cohomology exclusively with differential forms. In this work, Deligne cohomology is studied over the rationals, which introduces an obstruction to applying Deninger's results. The obstruction arises from the fact that the integration map from the de Rham complex to the Eilenberg-MacLane complex of the modular group is not an algebra homomorphism. We compute the correction terms of the integration map as regularized iterated integrals of Eisenstein series, and show that these integrals arise in the cup product and Massey triple product on Deligne cohomology.