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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

References

1. Atkin, A. O. L., Weierstrass points at cusps of T0(n), Ann. of Math. 85 (1967), 42–45.Google Scholar
2. Bateman, P., On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70–101.Google Scholar
3. Chevalley, C. and Weil, A., Über das Verhalten der Intégrale I. Gattung bei Automorphismen des Funktionenkôrpers, Abh. Math. Sem. Univ. Hamburg 10 (1934), 358–361.Google Scholar
4. Fricke, R., Lehrbuch der Algebra, Band 3 (Braunschweig, 1928).Google Scholar
5. Lehner, J. and Newman, M., Weierstrass points of Γ0(n), Ann. of Math. 79 (1964), 360–368.Google Scholar
6. Lewittes, J., Automorphisms of compact Riemann surfaces, Amer. J. Math. 85 (1963), 734–752.Google Scholar
7. Gaps at Weierstrass points for the modular group, Bull. Amer. Math. Soc. 69 (1963), 578–582.Google Scholar
8. Schoenberg, B., Uber die Weierstrass-Punkte in den Körpern der elliptischen Modulfunktionen, Abh. Math. Sem. Univ. Hamburg 17 (1951), 104–111.Google Scholar
9. Siegel, C. L., tfber die Klassenzahl quadratischer Zahlkörper, Acta Arith. 1 (1936), 83–86.Google Scholar
10. Smart, J. R., On Weierstrass points in the theory of elliptic modular forms, Math. Z. 94 (1966), 207–218.Google Scholar