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Hyperelliptic modular curves. (English) Zbl 0771.14008
For a positive integer $$N$$ and a subgroup $$\Delta$$ of the multiplicative group $$(\mathbb{Z}/N\mathbb{Z})^*$$ containing $$-1$$, let $$X_ \Delta(N)$$ denote the modular curve, defined over $$\mathbb{Q}$$, which corresponds to the modular group $\Gamma_ \Delta(N)=\left\{\left({a\;b\atop c\;d}\right)\in SL_ 2(\mathbb{Z});\;c\equiv 0\pmod N, (a\bmod N)\in\Delta\right\}.$ Specifically, for $$\Delta=\{\pm 1\}$$ (resp. $$\Delta=(\mathbb{Z}/N\mathbb{Z})^*)$$ one obtains the modular curves $$X_ 1(N)$$ (resp. $$X_ 0(N))$$ classifying pairs $$(E,P)$$ (resp. $$(E,\langle P\rangle))$$ of elliptic curves $$E$$ with a point $$P$$ (resp. cyclic group $$\langle P\rangle)$$ of order $$N$$ over the field of definition. The authors determine all hyperelliptic modular curves $$X_ \Delta(N)$$ of genus $$g_ \Delta(N)\geq 2$$. In fact they prove the following theorem:
The hyperelliptic modular curves $$X_ \Delta(N)$$ are the modular curves (of genus $$g_ 0(N)\geq 2)$$ $$X_ 0(N)$$ for $$N=22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 46, 47, 48, 50, 59, 71$$ and the modular curves (of genus $$g_ 1(N)=2)$$ $$X_ 1(N)$$ for $$N=13, 16, 18$$. – The associated hyperelliptic involutions are also given.
The proof is based on the fact that $$X_ \Delta(N)$$ is a subcovering of the covering $$X_ 1(N)\to X_ 0(N)$$, i.e. that we have a Galois covering $X_ 1(N)\to X_ \Delta(N)\to X_ 0(N): \quad (E,\pm P)\mapsto (E,\Delta P)\mapsto (E,\langle P\rangle).$ Use is made of work of A. P. Ogg [Bull. Soc. Math. Fr. 102(1974), 449-462 (1974; Zbl 0314.10018)] who determined all hyperelliptic modular curves $$X_ 0(N)$$.
The automorphism groups of the hyperelliptic modular curves $$X_ \Delta(N)$$ can also be given. Indeed, for square-free integers $$N$$, the automorphism groups of the modular curves $$X_ 0(N)$$ were determined by A. P. Ogg [Math. Ann. 228, 279-292 (1977; Zbl 0336.14002)], and the automorphism groups of $$X_ 1(N)$$ for square-free $$N$$ were found by the second author. In the present paper, the automorphism groups of the modular curves $$X_ 1(16)$$ and $$X_ 1(18)$$ are determined. It is shown that they are made up of $$2\times 2$$ matrices.
The above theorem is used for computing all torsion points on elliptic curves over quadratic fields. In fact, the conjecture of M. A. Kenku and F. Momose [Nagoya Math. J. 109, 125-149 (1988; Zbl 0647.14020)] above the only possible torsion groups of elliptic curves over quadratic fields turns out to be true due to work of S. Kamienny [Invent. Math. 109, No. 2, 221-229 (1992)].

##### MSC:
 14H52 Elliptic curves 14G35 Modular and Shimura varieties 14H10 Families, moduli of curves (algebraic) 14H30 Coverings of curves, fundamental group
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