Elliptic curves from sextics

Abstract

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, ${\mathcal{N}}_{\mathrm{torus}}/G$ and ${\mathcal{N}}_{\mathrm{gen}}/G$. 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 ${\mathcal{N}}_{\epsilon }/G$, torus, gen and their structure of elliptic fibration. Then we study the Mordell-Weil torsion groups of cubic curves $C_s$ over $Q$ and $D_s$ over $Q\left(\sqrt{-3}\right)$ respectively. We show that $C_s$ has the torsion group $Z/3Z$ for a generic $s\in Q$ and it also contains subfamilies which coincide with the universal families given by Kubert [Ku] with the torsion groups $Z/6Z,$$Z/6Z+Z/2Z,$$Z/9Z$, or $Z/12Z$. The cubic curves $D_s$ has torsion $Z/3Z+Z/3Z$ generically but also $Z/3Z+Z/6Z$ for a subfamily which is parametrized by $Q\left(\sqrt{-3}\right)$.