Equations of hyperelliptic modular curves
Annales de l'Institut Fourier, Volume 41 (1991) no. 4, pp. 779-795.

We compute, in a unified way, the equations of all hyperelliptic modular curves. The main tool is provided by a class of modular functions introduced by Newman in 1957. The method uses the action of the hyperelliptic involution on the cusps.

Nous calculons, avec une même méthode, les équations de toutes les courbes modulaires hyperelliptiques. L’outil principal est fourni par une classe de fonctions modulaires introduite par Newman en 1957. Notre méthode utilise l’action de l’involution hyperelliptique sur les pointes.

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     title = {Equations of hyperelliptic modular curves},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Institut Fourier},
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Rovira, Josep Gonzalez. Equations of hyperelliptic modular curves. Annales de l'Institut Fourier, Volume 41 (1991) no. 4, pp. 779-795. doi : 10.5802/aif.1273. http://archive.numdam.org/articles/10.5802/aif.1273/

[ALe] A. O. Atkin, J. Lehner, Hecke Operators on Г0(m), Math. Ann., 185 (1970), 134-160. | MR | Zbl

[B] B. J. Birch, Some calculations of modulars relations, in "Modular Functions of One Variable I", Springer Lecture Notes, 320, 175-186. | MR | Zbl

[F] R. Fricke, "Die elliptischen Funktionen und ihre Anwendungen, II", Teubner (1922). | JFM

[K] P. G. Kluit, On the Normalizer of Г0(N), in "Modular Functions of One Variable V". Springer Lecture Notes 601. | MR | Zbl

[LeN] J. Lehner, M. Newman, Weierstrass points of Г0(N), Ann. of Math., 79 (1964), 360-368. | MR | Zbl

[Li] G. Ligozat, Courbes modulaires de genre 1, Bull. Soc. Math. France, Mémoire, 43 (1975). | Numdam | MR | Zbl

[MS] B. Mazur, Swinnerton-Dyer, P : Arithmetic of Weil Curves, Inventiones Math., 25 (1974), 1-61. | Zbl

[N1] M. Newman, Construction and application of a class of modular functions, Proceed. of London Math. Soc., (1957), 334-350. | MR | Zbl

[N2] M. Newman, Construction and application of a class of modular functions (II), Proceed. of London Math. Soc., (1959), 373-387. | MR | Zbl

[O1] A. P. Ogg, Hyperelliptic Modular Curves, Bull. Soc. Math. France, 102 (1974), 449-462. | EuDML | Numdam | MR | Zbl

[O2] A. P. Ogg, On the Weierstarss points of X0(N), Illinois J. of Math., Vol. 22 (1978), 31-35. | MR | Zbl

[R] E. Reyssat, Quelques Aspects des Surfaces de Riemann, Birkhäusser, 1989. | MR | Zbl

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