Publication Date:
2015-01-17
Description:
We produce new explicit examples of genus- $2$ curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus- $2$ curves over ${{\mathbb {Q}}}$ whose Jacobians have a rational point of order $48$ , parameterized by a rank- $2$ elliptic curve over ${{\mathbb {Q}}}$ , and we exhibit a single genus- $2$ curve over ${{\mathbb {Q}}}$ whose Jacobian has a rational point of order $70$ , the largest order known. We also give new examples of genus- $2$ Jacobians with rational points of order $27$ , $28$ , and $39$ . Most of our examples are produced by ‘gluing’ two elliptic curves together along their $n$ -torsion subgroups, where $n$ is either $2$ or $3$ . The $2$ -gluing examples arise from techniques developed by the author in joint work with Leprévost and Poonen 15 years ago. The $3$ -gluing examples are made possible by an algorithm for explicit $3$ -gluing over non-algebraically closed fields recently developed by the author in joint work with Bröker, Lauter, and Stevenhagen.
Print ISSN:
0024-6093
Electronic ISSN:
1469-2120
Topics:
Mathematics
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