## Rational torsion points of some Jacobians of curves of genus 2. (Points rationnels de torsion de jacobiennes de certaines courbes de genre 2.)(French)Zbl 0783.14016

The author proves
(1) Let $$\ell$$ be one of the numbers 21, 22, 23, 25, 26, 27 and 29. Then there exists a curve $$C_ \ell$$ of genus 2, the Jacobian of which has a rational torsion point of order $$\ell$$. For $$\ell=24$$, the author constructs two curves with the same property – which curves are not $$\overline \mathbb{Q}$$-isomorphic (see theorem 1.1).
(2) There exists a family of curves of genus 2 defined over $$\mathbb{Q}$$, and depending on one parameter, such that its Jacobian has a subgroup isomorphic to $$\mathbb{Z}/3 \mathbb{Z} \times \mathbb{Z}/9 \mathbb{Z}$$ (see theorem 1.2).
Reviewer: A.Iliev (Sofia)

### MSC:

 14H40 Jacobians, Prym varieties 14G05 Rational points

### Keywords:

rational torsion point; Jacobian