Oka, Mutsuo Elliptic curves from sextics. (English) Zbl 1060.14035 J. Math. Soc. Japan 54, No. 2, 349-371 (2002). 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 \(\mathbb{Q}(\sqrt{-3})\). Cited in 1 Document MSC: 14H10 Families, moduli of curves (algebraic) 14H52 Elliptic curves Keywords:moduli space; Mordell-Weil torsion groups Citations:Zbl 0331.14010 PDFBibTeX XMLCite \textit{M. Oka}, J. Math. Soc. Japan 54, No. 2, 349--371 (2002; Zbl 1060.14035) Full Text: DOI arXiv