We give upper bounds for the maximal order of vanishing at ∞ of a modular form or cusp form of weight k on Γ0(Np) when p 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 ∈ S2(Γ0(Np) ) with ord∞(f) larger than the genus of X0(Np). In particular, this gives a (non-geometric) proof of a theorem of Ogg, which asserts that ∞ is not a Weierstrass point on X0(Np) if p N and X0(N) has genus zero.
ASJC Scopus subject areas
- Applied Mathematics