Equations of hyperelliptic modular curves. (English) Zbl 0758.14010
From the introduction: Several equations for the modular curves \(X_ 0(N)\) are known in the literature. They cover all values of \(N\) for which \(X_ 0(N)\) is elliptic and some of the values of \(N\) for which \(X_ 0(N)\) is hyperelliptic. Our goal is to give a procedure to compute, in a unified way, the equations of all hyperelliptic modular curves with \(g>0\). Our method uses some results of M. Newman [Proc. Lond. Math. Soc., III. Ser. 7, 334–350 (1957; Zbl 0097.28701); ibid. 9, 373–387 (1959; Zbl 0178.43001)] and ideas appearing in Birch’s computation of an equation for \(X_ 0(50)\) [B. J. Birch, in: Modular Functions one Variable I, Proc. Int. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 320, 175–186 (1973; Zbl 0261.10019)].
MSC:
| 14G35 | Modular and Shimura varieties |
| 14H25 | Arithmetic ground fields for curves |
| 11F11 | Holomorphic modular forms of integral weight |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
References:
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