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Weierstrass points at Cusps on special modular curves

Weierstrass points at Cusps on special modular curves Abh. Math. Sem. Univ. Hamburg 73 (2003), 241-251 By W. KOHNEN 1 Introduction Recall that a point x on a compact Riemann surface X of genus g is called a Weier- strass point if there exists a non-zero holomorphic differential form on X which vanishes at x of order > g. As is well-known, these points carry a lot of information on X, and there is a vast literature on the subject. Let M be the complex upper half-plane. For a positive integer N, let ro(N):{(~)~SLz(Z)[c-O (mod N)} be the Hecke congruence subgroup of level N which operates on ~ U ~1 (Q) in the usual way. Let Xo(N) be the complete modular curve of level N whose complex points can be identified with the quotient space F0(N)\~ U ~1 (Q) obtained from compactifying the curve F0(N)\o~ by adding the cusps. The curves Xo(N) play an important role in arithmetic, mainly because they classify N-isogeneous generalized elliptic curves. We let g(N) be the genus of Xo(N). Since q = e 2~riz (z c Jr is a local parameter at the cusp oo on Xo(N) and the map f(z) w-~ f(z)dz (z ~ ~) gives a natural identification of the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Weierstrass points at Cusps on special modular curves

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References (11)

Publisher
Springer Journals
Copyright
Copyright © 2003 by Mathematische Seminar
Subject
Mathematics; Algebra; Differential Geometry; Combinatorics; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02941280
Publisher site
See Article on Publisher Site

Abstract

Abh. Math. Sem. Univ. Hamburg 73 (2003), 241-251 By W. KOHNEN 1 Introduction Recall that a point x on a compact Riemann surface X of genus g is called a Weier- strass point if there exists a non-zero holomorphic differential form on X which vanishes at x of order > g. As is well-known, these points carry a lot of information on X, and there is a vast literature on the subject. Let M be the complex upper half-plane. For a positive integer N, let ro(N):{(~)~SLz(Z)[c-O (mod N)} be the Hecke congruence subgroup of level N which operates on ~ U ~1 (Q) in the usual way. Let Xo(N) be the complete modular curve of level N whose complex points can be identified with the quotient space F0(N)\~ U ~1 (Q) obtained from compactifying the curve F0(N)\o~ by adding the cusps. The curves Xo(N) play an important role in arithmetic, mainly because they classify N-isogeneous generalized elliptic curves. We let g(N) be the genus of Xo(N). Since q = e 2~riz (z c Jr is a local parameter at the cusp oo on Xo(N) and the map f(z) w-~ f(z)dz (z ~ ~) gives a natural identification of the

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Aug 28, 2008

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