Elliptic curves from sextics

Abstract

Let \mathscr{N} be the moduli space of sextics with 3 (3, 4)-cusps. The quotient moduli space \mathscr{N}/G is one-dimensional and consists of two components, \mathscr{N}_torus/G and \mathscr{N}_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 \mathscr{N}_ε/G, ε=torus, \ gen and their structure of elliptic fibration. Then we study the Mordell-Weil torsion groups of cubic curves C_s over \bm{Q} and D_s over \bm{Q}(√{-3}) respectively. We show that C_s has the torsion group \bm{Z}/3\bm{Z} for a generic s∈ \bm{Q} and it also contains subfamilies which coincide with the universal families given by Kubert [{Ku}] with the torsion groups \bm{Z}/6\bm{Z}, \bm{Z}/6\bm{Z}+\bm{Z}/2\bm{Z}, \bm{Z}/9\bm{Z}, or \bm{Z}/12\bm{Z}. The cubic curves D_s has torsion \bm{Z}/3\bm{Z}+\bm{Z}/3\bm{Z} generically but also \bm{Z}/3\bm{Z}+\bm{Z}/6\bm{Z} for a subfamily which is parametrized by \bm{Q}(√{-3}).