Holomorphic automorphic forms and cohomology

Bruggeman, Roelof; Choie, Young Ju; Diamantis, Nikolaos

Abstract

© 2018 by the American Mathematical Society. All rights reserved. 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.

Citation

Bruggeman, R., Choie, Y. J., & Diamantis, N. (2018). Holomorphic automorphic forms and cohomology. Memoirs of the American Mathematical Society, 253(1212), 1-182. https://doi.org/10.1090/memo/1212

• Journal Article Type: Article
• Acceptance Date: Dec 22, 2015
• Online Publication Date: Mar 29, 2018
• Publication Date: May 1, 2018
• Deposit Date: Apr 25, 2018
• Publicly Available Date: Apr 25, 2018
• Journal: Memoirs of the American Mathematical Society
• Print ISSN: 0065-9266
• Electronic ISSN: 1947-6221
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 253
• Issue: 1212
• Pages: 1-182
• DOI: https://doi.org/10.1090/memo/1212
• Keywords: Holomorphic automorphic form; Eichler integral; cohomology; mixed parabolic cohomology; period function; harmonic lift; harmonic functions; boundary germ
• Public URL: https://nottingham-repository.worktribe.com/output/922462
• Publisher URL: http://www.ams.org/books/memo/1212/
• Additional Information: First published in Memoirs of the American Mathematical Society in v. 253, no. 1212, 2018, published by the American Mathematical Society.
• Contract Date: Apr 25, 2018