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Journal of the London Mathematical Society (1)

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WEIERSTRASS POINTS ON $X_0(\lowercase{p}M)$ AND SUPERSINGULAR $\lowercase{j}$-INVARIANTS
AHMAD EL-GUINDY
Journal:

Journal of the London Mathematical Society / Volume 70 / Issue 1 / August 2004

Published online by Cambridge University Press:

23 July 2004, pp. 1-22

Print publication:

August 2004
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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 $p\nmid M$), and he proved that if $Q$ is a $\mathbb{Q}$-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.