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Vanishing of modular forms at infinity

Authors: Scott Ahlgren, Nadia Masri and Jeremy Rouse
Journal: Proc. Amer. Math. Soc. 137 (2009), 1205-1214
MSC (2000): Primary 11F11, 11F33, 14H55
Published electronically: November 21, 2008
MathSciNet review: 2465641
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Abstract: We give upper bounds for the maximal order of vanishing at $ \infty$ of a modular form or cusp form of weight $ k$ on $ \Gamma_0(Np)$ when $ p\nmid N$ is prime. The results improve the upper bound given by the classical valence formula and the bound (in characteristic $ p$) given by a theorem of Sturm. In many cases the bounds are sharp. As a corollary, we obtain a necessary condition for the existence of a non-zero form $ f\in S_2(\Gamma_0(Np))$ with $ \operatorname{ord} _\infty(f)$ larger than the genus of $ X_0(Np)$. In particular, this gives a (non-geometric) proof of a theorem of Ogg, which asserts that $ \infty$ is not a Weierstrass point on $ X_0(Np)$ if $ p\nmid N$ and $ X_0(N)$ has genus zero.

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Additional Information

Scott Ahlgren
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Nadia Masri
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Jeremy Rouse
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Received by editor(s): April 9, 2008
Published electronically: November 21, 2008
Additional Notes: The first author thanks the National Science Foundation for its support through grant DMS 01-34577.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.