Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Abstract

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X0(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X0(N), if N=pM with p prime, p∤M and the genus of X0(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.