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Elliptic curves from sextics. (English) Zbl 1060.14035

Summary: Let \({\mathcal N}\) be the moduli space of sextics with 3 \((3,4)\)-cusps. The quotient moduli space \({\mathcal N}/G\) is one-dimensional and consists of two components, \({\mathcal N}_{\text{torus} }/G\) and \({\mathcal N}_{\text{gen}}/G\). By quadratic transformations, they are transformed into one-parameter families \(C_s\) and \(D_s\) of cubic curves respectively. First we study the geometry of \({\mathcal N}_{\text{torus}}/G\), and \({\mathcal N}_{\text{gen}}/G\) and their structure of the elliptic fibration. Then we study the Mordell-Weil torsion groups of cubic curves \(C_s\) over \(\mathbb{Q}\) and \(D_s\) over \(\mathbb{Q}(\sqrt{-3})\) respectively. We show that \(C_s\) has the torsion group \(\mathbb{Z}/3\mathbb{Z}\) for a generic \(s\in \mathbb{Q}\) and it also contains subfamilies which coincide with the universal families given by D. S. Kubert [Proc. London Math. Soc. (3), 33, 193–237 (1976; Zbl 0331.14010)] with the torsion groups \(\mathbb{Z}/6 \mathbb{Z}\), \(\mathbb{Z}/6\mathbb{Z}+\mathbb{Z}/2 \mathbb{Z}\), \(\mathbb{Z}/9\mathbb{Z}\), or \(\mathbb{Z}/12 \mathbb{Z}\). The cubic curves \(D_s\) has torsion \(\mathbb{Z}/3\mathbb{Z}+\mathbb{Z}/3\mathbb{Z}\) generically but also \(\mathbb{Z}/3\mathbb{Z}+ \mathbb{Z}/6\mathbb{Z}\) for a subfamily which is parametrized by \(\mathbb{Q}(\sqrt{-3})\).

MSC:

14H10 Families, moduli of curves (algebraic)
14H52 Elliptic curves

Citations:

Zbl 0331.14010
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