Weierstrass points on certain modular groups

Abstract

We investigate Weierstrass points of the modular curve X-Delta(N) of genus >= 2 when Delta is a proper subgroup of (Z/NZ)*. Let N = p(2)M where p is a prime number and M is a positive integer. Modifying Atkin's method in the case +/-(1 + pM) is an element of A, we find conditions for the cusp 0 to be a Weierstrass point on the modular curve X-Delta(p(2)M). Moreover, applying Schoneberg's theorem we show that except for finitely many N, the fixed points of the Fricke involutions W-N are Weierstrass points on X-Delta(N). (C) 2015 Elsevier Inc. All rights reserved