Holomorphic automorphic forms and cohomology
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...
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| Language: | English |
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American Mathematical Society
2018
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| Online Access: | https://eprints.nottingham.ac.uk/41245/ |
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| author | Bruggeman, Roelof Choie, YoungJu Diamantis, Nikolaos |
| author_facet | Bruggeman, Roelof Choie, YoungJu Diamantis, Nikolaos |
| author_sort | Bruggeman, Roelof |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. Our result concerns arbitrary cofinite discrete groups with cusps, and 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. |
| first_indexed | 2025-11-14T19:44:40Z |
| format | Article |
| id | nottingham-41245 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T19:44:40Z |
| publishDate | 2018 |
| publisher | American Mathematical Society |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-412452018-04-25T08:47:08Z https://eprints.nottingham.ac.uk/41245/ Holomorphic automorphic forms and cohomology Bruggeman, Roelof Choie, YoungJu Diamantis, Nikolaos 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. Our result concerns arbitrary cofinite discrete groups with cusps, and 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. American Mathematical Society 2018-03-29 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/41245/1/1404.6718.pdf Bruggeman, Roelof, Choie, YoungJu and Diamantis, Nikolaos (2018) Holomorphic automorphic forms and cohomology. Memoirs of the American Mathematical Society, 253 (1212). ISSN 0065-9266 http://www.ams.org/books/memo/1212/ doi:10.1090/memo/1212 doi:10.1090/memo/1212 |
| spellingShingle | Bruggeman, Roelof Choie, YoungJu Diamantis, Nikolaos Holomorphic automorphic forms and cohomology |
| title | Holomorphic automorphic forms and cohomology |
| title_full | Holomorphic automorphic forms and cohomology |
| title_fullStr | Holomorphic automorphic forms and cohomology |
| title_full_unstemmed | Holomorphic automorphic forms and cohomology |
| title_short | Holomorphic automorphic forms and cohomology |
| title_sort | holomorphic automorphic forms and cohomology |
| url | https://eprints.nottingham.ac.uk/41245/ https://eprints.nottingham.ac.uk/41245/ https://eprints.nottingham.ac.uk/41245/ |