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The image and kernel of Atkin's $ U_{p}$ operator modulo $ p$


Author: Michael Dewar
Journal: Proc. Amer. Math. Soc. 140 (2012), 1931-1938
MSC (2010): Primary 11F11, 11F33
DOI: https://doi.org/10.1090/S0002-9939-2011-11115-8
Published electronically: October 14, 2011
MathSciNet review: 2888180
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Abstract: We compute the image of Atkin's $ U_{p}$ operator on reduced modular forms. If $ A\geq 1$ and $ 2\leq B \leq p+1$, then $ U_{p}: \widetilde M_{Ap+B}\left ( \Gamma _{1}(N) \right ) \onto \widetilde M_{A+B} \left (\Gamma _{1}(N) \right )$ is a surjection. In particular, the dimension of $ \ker U_{p}$ is known for weights at least $ p+2$.


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Additional Information

Michael Dewar
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada

DOI: https://doi.org/10.1090/S0002-9939-2011-11115-8
Received by editor(s): February 9, 2011
Published electronically: October 14, 2011
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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