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Modular cocycles and cup product

Title
Modular cocycles and cup product
Authors
Roelof BruggemanYoungJu Choie
Date Issued
31-Jul-2019
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Abstract
The Eichler-Shimura isomorphism relates holomorphic modular cusps forms of positive even integral weight to cohomology classes. The Haberland formula uses the cup product to give a cohomological formulation of the Petersson scalar product. In this paper we extend Haberland's formula to modular cusp forms of positive real weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We may also consider the cup product of two Eichler cocycles. In the classical situation this cup product is almost always zero. However we show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by duality with a space of entire modular forms. The cup product yields a bilinear map over C from pairs of holomorphic modular forms (not necessarily of the same weight, one of them may have large growth at the cusps) to coinvariants in infinite-dimensional modules. To investigate whether this bilinear map is non-trivial we test the result against entire modular forms of a suitable weight. Under some conditions on the weights, this leads to an explicit triple integral, which can be investigated numerically, thus providing evidence that the cup product is non-trivial at least in some situations. (C) 2019 Elsevier Inc. All rights reserved.
Keywords
EICHLER COHOMOLOGY; FORMS; PERIODS
URI
http://oasis.postech.ac.kr/handle/2014.oak/98895
ISSN
0001-8708
Article Type
Article
Citation
ADVANCES IN MATHEMATICS, vol. 351, page. 296 - 342, 2019-07-31
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 CHOIE, YOUNG JU
Dept of Mathematics
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