Genus-2 Jacobians with torsion points of large order.

*(English)*Zbl 1319.14030This short and clear article is concerned with the explicit construction of smooth curves of genus \(2\) over \(\mathbb{Q}\) whose Jacobian varieties admit a torsion point of large order. Among the curves constructed is one whose Jacobian admits a torsion point over \(\mathbb{Q}\) of order \(70\); moreover, the author obtains infinitely many curves whose Jacobians all have a torsion point of order \(48\).

Two main methods are used to construct the curves in question. The first of these uses techniques developed in [R. Bröker et al., LMS J. Comput. Math. 18, 170–197 (2015; Zbl 1387.14086)]: given two elliptic curves \(E_1\) and \(E_2\) along with an isomorphism \(\psi : E_1 [3] \to E_2 [3]\) that is an anti-isometry with respect to the Weil pairings, the quotient of \(E_1 \times E_2\) by the graph of \(\psi\) is a principally polarized abelian variety in a natural way, and if this is indeed the Jacobian of a smooth curve of genus \(2\), then an equation of such a curve can be effectively determined. Using two elliptic curves over \(\mathbb{Q}\) with torsion points of order \(7\) and \(10\), respectively, this construction yields a curve of genus \(2\) whose Jacobian admits a torsion point of order \(70\) over the same field.

The aforementioned infinite family is constructed by using \(2\)-torsion instead of \(3\)-torsion in the argument above. In this case, the corresponding “gluing techniques” were developed in [E. W. Howe et al., Forum Math. 12, No. 3, 315–364 (2000; Zbl 0983.11037)]. In his application, the author first fixes a curve \(E_1\) with an \(8\)-torsion point and shows that it can be glued with all curves \(E_2\) in a certain family that depend rationally on \(2\) parameters and whose members all admit a \(6\)-torsion point. The gluing techniques then yield a family of curves genus \(2\) along with a torsion point of order \(24\) on their Jacobian, which is parametrized by a rational variety \(V\) defined over \(\mathbb{Q}\). The subtlety lies in showing that for infinitely many points \(v\) in \(V (\mathbb{Q})\) the torsion point on the corresponding Jacobian is actually the double of another \(\mathbb{Q}\)-rational point. In geometric terms, the author accomplishes this by showing that there exists a map of surfaces \(U \to V\) defined over \(\mathbb{Q}\) such that \(v\) is such a double if and only if it lies in the image of \(U (\mathbb{Q})\), and subsequently exhibiting a curve of genus \(1\) on \(U\) with infinitely many \(\mathbb{Q}\)-rational points.

The article concludes by giving new examples of curves of genus \(2\) that allow a defining equation with small coefficients, yet whose Jacobians still admit a torsion point of large order. These were found by a computer search.

Two main methods are used to construct the curves in question. The first of these uses techniques developed in [R. Bröker et al., LMS J. Comput. Math. 18, 170–197 (2015; Zbl 1387.14086)]: given two elliptic curves \(E_1\) and \(E_2\) along with an isomorphism \(\psi : E_1 [3] \to E_2 [3]\) that is an anti-isometry with respect to the Weil pairings, the quotient of \(E_1 \times E_2\) by the graph of \(\psi\) is a principally polarized abelian variety in a natural way, and if this is indeed the Jacobian of a smooth curve of genus \(2\), then an equation of such a curve can be effectively determined. Using two elliptic curves over \(\mathbb{Q}\) with torsion points of order \(7\) and \(10\), respectively, this construction yields a curve of genus \(2\) whose Jacobian admits a torsion point of order \(70\) over the same field.

The aforementioned infinite family is constructed by using \(2\)-torsion instead of \(3\)-torsion in the argument above. In this case, the corresponding “gluing techniques” were developed in [E. W. Howe et al., Forum Math. 12, No. 3, 315–364 (2000; Zbl 0983.11037)]. In his application, the author first fixes a curve \(E_1\) with an \(8\)-torsion point and shows that it can be glued with all curves \(E_2\) in a certain family that depend rationally on \(2\) parameters and whose members all admit a \(6\)-torsion point. The gluing techniques then yield a family of curves genus \(2\) along with a torsion point of order \(24\) on their Jacobian, which is parametrized by a rational variety \(V\) defined over \(\mathbb{Q}\). The subtlety lies in showing that for infinitely many points \(v\) in \(V (\mathbb{Q})\) the torsion point on the corresponding Jacobian is actually the double of another \(\mathbb{Q}\)-rational point. In geometric terms, the author accomplishes this by showing that there exists a map of surfaces \(U \to V\) defined over \(\mathbb{Q}\) such that \(v\) is such a double if and only if it lies in the image of \(U (\mathbb{Q})\), and subsequently exhibiting a curve of genus \(1\) on \(U\) with infinitely many \(\mathbb{Q}\)-rational points.

The article concludes by giving new examples of curves of genus \(2\) that allow a defining equation with small coefficients, yet whose Jacobians still admit a torsion point of large order. These were found by a computer search.

Reviewer: Jeroen Sijsling (Hanover)

##### MSC:

14G05 | Rational points |

11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |

14H25 | Arithmetic ground fields for curves |

14H45 | Special algebraic curves and curves of low genus |

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