TY - BOOK ID - 147838346 TI - Period functions for Maass wave forms and cohomology AU - Bruggeman, Roelof W. AU - Lewis, J. AU - Zagier, Don PY - 2015 SN - 1470425033 PB - Providence, Rhode Island : American Mathematical Society, DB - UniCat KW - Forms, Modular. KW - Forms (Mathematics) KW - Cohomology operations. KW - Algebraic topology. UR - https://www.unicat.be/uniCat?func=search&query=sysid:147838346 AB - The authors construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups Gammasubsetmathrm{PSL}_2({mathbb{R}}). In the case that Gamma is the modular group mathrm{PSL}_2({mathbb{Z}}) this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191-258, where a bijection was given between cuspidal Maass forms and period functions. The authors introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables them to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all Gamma-invariant eigenfunctions of the Laplace operator. For spaces of Maass cusp forms the authors also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. They use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology. ER -