### 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 X_{0}(M) has genus zero and p prime with p ł 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.

Original language | English |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Journal of the London Mathematical Society |

Volume | 70 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 2004 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Weierstrass points on X _{0}(pM) and supersingular j-invariants.** / ElGuindy, Ahmad.

Research output: Contribution to journal › Article

_{0}(pM) and supersingular j-invariants',

*Journal of the London Mathematical Society*, vol. 70, no. 1, pp. 1-22. https://doi.org/10.1112/S0024610704005496

}

TY - JOUR

T1 - Weierstrass points on X0(pM) and supersingular j-invariants

AU - ElGuindy, Ahmad

PY - 2004/8

Y1 - 2004/8

N2 - 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 ł 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.

AB - 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 ł 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.

UR - http://www.scopus.com/inward/record.url?scp=4344669363&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344669363&partnerID=8YFLogxK

U2 - 10.1112/S0024610704005496

DO - 10.1112/S0024610704005496

M3 - Article

AN - SCOPUS:4344669363

VL - 70

SP - 1

EP - 22

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 1

ER -