Large torsion subgroups of split Jacobians of curves of genus two or three

1998/09/08 by Everett W. Howe, Franck Leprevost, Bjorn Poonen · 1 citation
Mathematics · #math.NT

paper · pdf

published as Forum Math. 12, no. 3 (2000), 315--364

arxiv created 1998/09/08 · arxiv updated 2009/12/01

Abstract

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P¹ of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X⁴ + Y⁴ + Z⁴) - 96914(X² Y² + X² Z² + Y² Z²) = 0, whose Jacobian has 864 rational torsion points. This paper has appeared in Forum Math. 12 (2000) 315-364.

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• Curves of genus two with maps of every degree to a fixed elliptic curve

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