Weierstrass points on certain modular groups

Abstract

We investigate Weierstrass points of the modular curve XΔ(N) of genus ≥2 when Δ is a proper subgroup of (Z/NZ)⁎. Let N=p2M where p is a prime number and M is a positive integer. Modifying Atkin's method in the case ±(1+pM)∈Δ, we find conditions for the cusp 0 to be a Weierstrass point on the modular curve XΔ(p2M). Moreover, applying Schöneberg's theorem we show that except for finitely many N, the fixed points of the Fricke involutions WN are Weierstrass points on XΔ(N).