Vanishing of modular forms at infinity

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.