dc.contributor.author Bruggeman, Roelof W. dc.contributor.author Ch'oe, Yŏng-ju dc.contributor.author Diamantis, Nikolaos dc.date.accessioned 2014-04-25T12:00:02Z dc.date.accessioned 2016-10-05T14:13:58Z dc.date.available 2014-04-25T12:00:02Z dc.date.available 2016-10-05T14:13:58Z dc.date.issued 2014-04-25 dc.identifier.uri http://publications.mfo.de/handle/mfo/1078 dc.description Research in Pairs 2014 en_US dc.description.abstract We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. We use Knopp’s generalization of this integral to real weights, and apply it to complex weights that are not an integer at least 2. We show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. We impose no condition on the growth of the automorphic forms at the cusps. So our result covers exponentially growing automorphic forms, like those studied by Borcherds, and like those in the theory of mock automorphic forms. For real weights that are not an integer at least 2 we similarly characterize the space of cusp forms and the space of entire automorphic forms. We give a relation between the cohomology classes attached to holomorphic automorphic forms of real weight and the existence of harmonic lifts. A tool in establishing these results is the relation to cohomology groups with values in modules of “analytic boundary germs”, which are represented by harmonic functions on subsets of the upper half-plane. It turns out that for integral weights at least 2 the map from general holomorphic automorphic forms to cohomology with values in analytic boundary germs is injective. So cohomology with these coefficients can distinguish all holomorphic automorphic forms, unlike the classical Eichler theory. en_US dc.language.iso en en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2014,07 dc.subject Holomorphic automorphic form en_US dc.subject Eichler integral en_US dc.subject Cohomology en_US dc.subject Mixed parabolic cohomology en_US dc.subject Period function en_US dc.subject Harmonic lift en_US dc.subject Harmonic functions en_US dc.subject Boundary germ en_US dc.title Holomorphic automorphic forms and cohomology en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2014-07 local.scientificprogram Research in Pairs 2014 local.series.id OWP-2014-07 local.subject.msc 11 local.subject.msc 22
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