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Genus-2 Jacobians with torsion points of large order. (English) Zbl 1319.14030
This 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.

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
Full Text: DOI arXiv
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