Vanishing of modular forms at infinity

Ahlgren, Scott; Masri, Nadia; Rouse, Jeremy

doi:10.1090/S0002-9939-08-09768-2

Vanishing of modular forms at infinity
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by Scott Ahlgren, Nadia Masri and Jeremy Rouse

Proc. Amer. Math. Soc. 137 (2009), 1205-1214

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

Published electronically: November 21, 2008

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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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