Elliptic curves from sextics

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Author(s)

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}<SUB>torus</SUB>/G and \mathscr{N}<SUB>gen</SUB>/G. By quadratic transformations, they are transformed into one-parameter 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 Mordell-Weil 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), 349-371, 2002-04

    The Mathematical Society of Japan

References:  19

Codes

  • NII Article ID (NAID)
    10008204985
  • NII NACSIS-CAT ID (NCID)
    AA0070177X
  • Text Lang
    ENG
  • Article Type
    ART
  • ISSN
    00255645
  • NDL Article ID
    6153496
  • NDL Source Classification
    ZM31(科学技術--数学)
  • NDL Call No.
    Z53-A209
  • Data Source
    CJP  NDL  J-STAGE 
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