Elliptic curves from sextics

 Oka Mutsuo OKA Mutsuo
 Department of Mathematics Tokyo Metropolitan University
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Author(s)

 Oka Mutsuo OKA Mutsuo
 Department of Mathematics Tokyo Metropolitan University
Abstract
Let \mathscr{N} be the moduli space of sextics with 3 (3, 4)cusps. The quotient moduli space \mathscr{N}/G is onedimensional and consists of two components, \mathscr{N}<SUB>torus</SUB>/G and \mathscr{N}<SUB>gen</SUB>/G. By quadratic transformations, they are transformed into oneparameter families C<SUB>s</SUB> and D<SUB>s</SUB> of cubic curves respectively. First we study the geometry of \mathscr{N}<SUB>ε</SUB>/G, ε=torus, \ gen and their structure of elliptic fibration. Then we study the MordellWeil torsion groups of cubic curves C<SUB>s</SUB> over \bm{Q} and D<SUB>s</SUB> over \bm{Q}(√{3}) respectively. We show that C<SUB>s</SUB> 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<SUB>s</SUB> 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}).
Journal

 Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 54(2), 349371, 200204
The Mathematical Society of Japan
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