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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 growth of the Taylor coefficients of automorphic forms


Author: Thomas A. Metzger
Journal: Proc. Amer. Math. Soc. 39 (1973), 321-328
MSC: Primary 10D15
MathSciNet review: 0313193
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Abstract: The growth of the Taylor coefficients of an automorphic form of dimension -2 with respect to a Fuchsian group $ \Gamma $ is related to the area integral $ \smallint {\smallint _U}\vert F{\vert^s}{(1 - \vert z{\vert^2})^t}dxdy$, and it is found that these coefficients must grow faster than a power of $ n$. Moreover if $ F \in H(p,\Gamma )$ then these coefficients must grow slower than a different power of $ n$ and, in fact, $ {a_n}/n$ is square summable if either $ p = 2$ or $ 1 < p < \infty $ and $ \Gamma $ is finitely generated of the second kind.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0313193-5
PII: S 0002-9939(1973)0313193-5
Keywords: Taylor coefficients, $ p$-integrable, automorphic forms, Fuchsian groups
Article copyright: © Copyright 1973 American Mathematical Society