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Large torsion subgroups of split Jacobians of curves of genus two or three. (English) Zbl 0983.11037
Over twenty years ago, Mazur solved the problem of finding the groups that do occur as rational torsion subgroups of elliptic curves over \(\mathbb{Q}\). The analogous problem for Jacobian varieties of curves over \(\mathbb{Q}\) of genus \(>1\) is still open. Note that the largest known group of rational torsion has order 30 for a 2-dimensional Jacobian and order 64 for a 3-dimensional Jacobian.
The authors present two finite sets \(L_j\) \((j=1,2)\) of abstract groups such that, for every group \(G\) in \(L_j\), there exists a family of curves over \(\mathbb{Q}\) of genus \(j+1\) parametrized by the rational points on a non-empty Zariski-open subset of a variety, whose Jacobians contain a group of rational points isomorphic to \(G\). Especially, the authors obtain a family of genus 2 curves over \(\mathbb{Q}\) whose Jacobians each have 128 rational torsion points. Further, they find a genus 3 curve over \(\mathbb{Q}\) whose Jacobian has 864 rational torsion points.

MSC:
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H40 Jacobians, Prym varieties
11G10 Abelian varieties of dimension \(> 1\)
14H25 Arithmetic ground fields for curves
14H45 Special algebraic curves and curves of low genus
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