Weierstrass Points on X 0 (pM) and Supersingular j ‐Invariants

Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on X 0 (pM) (for M such that X 0 (M) has genus zero and p prime with p ∤ M ), and he proved that if Q is a Q‐rational Weierstrass point on X 0 (pM) , then its reduction modulo p is supersingular. The paper shows that, for square‐free M on the list, all supersingular j ‐invariants are obtained in this way. Furthermore, for most cases where M is prime, the explicit correspondence between Weierstrass points and supersingular j ‐invariants in characteristic p is described. Along the way, a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo p is generalized.