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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the Petersson scalar product of arbitrary modular forms


Authors: Vicentiu Pasol and Alexandru A. Popa
Journal: Proc. Amer. Math. Soc. 142 (2014), 753-760
MSC (2010): Primary 11F11
Published electronically: November 19, 2013
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Abstract: We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $ k\ge 2$ for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show that the Petersson product is nondegenerate for $ \Gamma _1(N)$ and $ k>2$. For $ k=2$ we give examples when it is degenerate and when it is nondegenerate.


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Additional Information

Vicentiu Pasol
Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: vicentiu.pasol@imar.ro

Alexandru A. Popa
Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: alexandru.popa@imar.ro

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11815-0
PII: S 0002-9939(2013)11815-0
Received by editor(s): April 3, 2012
Published electronically: November 19, 2013
Additional Notes: The first author was partially supported by the CNCSIS grant PD-171/28.07.2010
The second author was partially supported by the European Community grant PIRG05-GA-2009-248569
Communicated by: Ken Ono
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.