Vanishing of modular forms at infinity

Abstract

We give upper bounds for the maximal order of vanishing at ∞ \infty of a modular form or cusp form of weight k k on Γ 0 ( N p ) \Gamma _0(Np) when p ∤ N p\nmid N is prime. The results improve the upper bound given by the classical valence formula and the bound (in characteristic p 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 ∈ S 2 ( Γ 0 ( N p ) ) f\in S_2(\Gamma _0(Np)) with ord ∞ ⁡ ( f ) \operatorname {ord} _\infty (f) larger than the genus of X 0 ( N p ) 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 ( N p ) X_0(Np) if p ∤ N p\nmid N and X 0 ( N ) X_0(N) has genus zero.