WEIERSTRASS POINTS ON $X_0(\lowercase{p}M)$ AND SUPERSINGULAR $\lowercase{j}$-INVARIANTS

Abstract

Weierstrass points are special points on a Riemann surface that carry 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 on X0(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.