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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.

##### 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
##### Keywords:
curves of genus 2; torsion points; rational points
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##### References:
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