Weierstrass Points on X0(pM) and Supersingular j-Invariants

Weierstrass points are special points on a Riemann surface thatcarry a lot of information. Ogg studied such points on X₀(pM)(for M such that X₀(M) has genus zero and p prime with p M),and he proved that if Q is a Q-rational Weierstrass point onX₀(pM), then its reduction modulo p is supersingular. The papershows that, for square-free M on the list, all supersingularj-invariants are obtained in this way. Furthermore, for mostcases where M is prime, the explicit correspondence betweenWeierstrass points and supersingular j-invariants in characteristicp is described. Along the way, a useful formula of Rohrlichfor computing a certain Wronskian of modular forms modulo pis generalized.