Elliptic Curves from Sextics

1999/12/06 by Mutsuo Oka, Oka, Mutsuo
Mathematics · #14H20 #14H45 #14H52 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H20 #msc:14H45 #msc:14H52

paper · pdf · doi:10.48550/arxiv.math/9912041

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

arxiv created 1999/12/16 · arxiv updated 2016/09/07

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(√(-3)) respectively. We show that C_s has the torsion group \bf Z/3\bf Z for a generic s∈ \bf Q and it also contains subfamilies which coincide with the universal families given by Kubert with the torsion groups \bf Z/6\bf Z, \bf Z/6\bf Z+\bf Z/2\bf Z, \bf Z/9\bf Z or \bf Z/12\bf Z. The cubic curves D_s has torsion \bf Z/3\bf Z+\bf Z/3\bf Z generically but also \bf Z/3\bf Z+\bf Z/6\bf Z for a subfamily which is parametrized by \bf Q(√(-3)) .

Related

• Torsion groups of Mordell curves over cubic and sextic fields
• Torsion groups of Mordell curves over number fields of higher degree
• A geometric approach to elliptic curves with torsion groups ℤ/10ℤ, ℤ/12ℤ, ℤ/14ℤ, and ℤ/16ℤ
• Torsion of Elliptic Curves over Quadratic Fields
• A parametrised family of Mordell curves