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Proceedings of the American Mathematical Society

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Modular forms on Hecke's modular groups


Author: Ronald J. Evans
Journal: Proc. Amer. Math. Soc. 37 (1973), 411-412
MSC: Primary 10D05
DOI: https://doi.org/10.1090/S0002-9939-1973-0309872-6
MathSciNet review: 0309872
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Abstract: Let $ H = \{ \tau = x + iy:y > 0\} $. Let $ \lambda > 0,k > 0,\gamma = \pm 1$. Let $ M(\lambda ,k,\gamma )$ denote the set of functions f for which $ f(\tau ) = \sum _{n = 0}^\infty {a_n}{e^{2\pi in\tau /\lambda }}$ and $ f( - 1/\tau ) = \gamma {(\tau /i)^k}f(\tau )$, for all $ \tau \in H$. Let $ {M_0}(\lambda ,k,\gamma )$ denote the set of $ f \in M(\lambda ,k.\gamma )$ for which $ f(\tau ) = O({y^c})$ uniformly for all x as $ y \to {0^ + }$, for some real c. We give a new proof that if $ \lambda = 2\cos (\pi /q)$ for an integer $ q \geqq 3$, then $ M(\lambda ,k,\gamma ) = {M_0}(\lambda ,k,\gamma )$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0309872-6
Keywords: Modular form, Hecke modular groups, fundamental region, equivalent points
Article copyright: © Copyright 1973 American Mathematical Society

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