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, ${\mathcal{N}}_{torus}/G$ and ${\mathcal{N}}_{gen}/G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. We study the Mordell-Weil torsion groups of cubic curves $C_s$ over $\bfQ$ and $D_s$ over $\bfQ(\sqrt{-3})$ respectively. We show that $C_s$ has the torsion group $Z/3\mathbf{Z}$ for a generic $s\in \mathbf{Q}$ and it also contains subfamilies which coincide with the universal families given by Kubert with the torsion groups $Z/6\mathbf{Z}$, $Z/6\mathbf{Z}+Z/2\mathbf{Z}$, $Z/9\mathbf{Z}$ or $Z/12\mathbf{Z}$. The cubic curves $D_s$ has torsion $Z/3\mathbf{Z}+Z/3\mathbf{Z}$ generically but also $Z/3\mathbf{Z}+Z/6\mathbf{Z}$ for a subfamily which is parametrized by $\mathbf{Q}(\sqrt{-3})$.

A remark is added. (after replace mistake). 16 pages