Ahmad El-Guindy, University of Wisconsin
Title: Weierstrass Points on X_0(pM) and Supersingular j-invariants
ABSTRACT: 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 the genus of X_0(M) is 0 and prime p not dividing M) and proved that the reduction of Weierstrass points on X_0(pM) is supersingular mod p. In this talk we show that, for square free M on the list, all supersingular j-invariants are uniformly covered this way. Furthermore, in most cases where M is prime we describe the explicit correspondence between Weierstrass points and supersingular j-invariants. Along the way we also generalize a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo p.