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Weierstrass Points at the Cusps of Γ0(16p) and Hyperellipticity of Γ0(n)

Published online by Cambridge University Press:  20 November 2018

H. Larcher
H. Larcher*
Affiliation:
University of Maryland, Munich Campus, Munich, Germany
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For a fixed positive integer n we consider the subgroup Γ0(n) of the modular group Γ(l). Γ0(n) consists of all linear fractional transformations L: z → (az + b)/(cz + d) with rational integers a, b, c, d, determinant ad – bc = 1, and c ≡ 0(mod n). If ℋ = {z|z = x + iy, x and y real and y > 0} is the upper half of the z-plane then S0 = S0(n) = ℋ/Γ0(n), properly compactified, is a compact Riemann surface whose genus we denote by g(n). A point P of a Riemann surface S of genus g is called a Weierstrass point if there exists a function on S that has a pole of order α ≦ g at P and is regular everywhere else on S.

Lehner and Newman started the search for Weierstrass points of S0 (or, loosely, of Γ0(n)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 6 , December 1971 , pp. 960 - 968
Copyright
Copyright © Canadian Mathematical Society 1971

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