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

DOI:
https://doi.org/10.1090/S0002-9939-08-09768-2

Published electronically:
November 21, 2008

MathSciNet review:
2465641

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Abstract | References | Similar Articles | Additional Information

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

Email:
ahlgren@math.uiuc.edu

**Nadia Masri**

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

Email:
nmasri@math.uiuc.edu

**Jeremy Rouse**

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

MR Author ID:
741123

Email:
jarouse@math.uiuc.edu

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.