Hyperelliptic modular curves
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 \textit{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 \textit{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 \textit{M. A. Kenku} and \textit{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 \textit{S. Kamienny} [Invent. Math. 109, No. 2, 221-229 (1992)].
automorphism groups of the hyperelliptic modular curves
torsion points of elliptic curves over quadratic fields
scientific article; zbMATH DE number 61126