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April, 2002 Elliptic curves from sextics
Mutsuo OKA
J. Math. Soc. Japan 54(2): 349-371 (April, 2002). DOI: 10.2969/jmsj/05420349


Let N be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space N/G is one-dimensional and consists of two components, Ntorus/G and Ngen/G. By quadratic transformations, they are transformed into one-parameter families Cs and Ds of cubic curves respectively. First we study the geometry of Nε/G, &= torus, gen and their structure of elliptic fibration. Then we study the Mordell-Weil torsion groups of cubic curves Cs over Q and Ds over Q(-3) respectively. We show that Cs has the torsion group Z/3Z for a generic sQ 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 Ds has torsion Z/3Z+Z/3Z generically but also Z/3Z+Z/6Z for a subfamily which is parametrized by Q(-3).


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Mutsuo OKA. "Elliptic curves from sextics." J. Math. Soc. Japan 54 (2) 349 - 371, April, 2002.


Published: April, 2002
First available in Project Euclid: 9 June 2008

zbMATH: 1060.14035
MathSciNet: MR1883523
Digital Object Identifier: 10.2969/jmsj/05420349

Primary: 14H10
Secondary: 14H20 , 14H52

Keywords: dual curves , Elliptic curves , Mordell-Weil torsion , Sextics

Rights: Copyright © 2002 Mathematical Society of Japan


Vol.54 • No. 2 • April, 2002
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