## Jacobians of genus-2 curves with a rational point of order 11.(English)Zbl 1244.11064

Summary: On the one hand, it is well known that Jacobians of (hyper)elliptic curves defined over $$\mathbb Q$$ having a rational point of order $$l$$ can be used in many applications, for instance in the construction of class groups of quadratic fields with a nontrivial $$l$$-rank. On the other hand, it is also well known that 11 is the least prime number that is not the order of a rational point of an elliptic curve defined over $$\mathbb Q$$. It is therefore interesting to look for curves of higher genus whose Jacobians have a rational point of order 11. This problem has already been addressed, and E. V. Flynn [J. Number Theory 36, No. 3, 257–265 (1990; Zbl 0757.14025)] found such a family $$\mathfrak F_t$$ of genus-2 curves. Now it turns out that the Jacobian $$J_0(23)$$ of the modular genus-2 curve $$X_0(23)$$ has the required property, but does not belong to $$\mathfrak F_t$$. The study of $$X_0(23)$$ leads to a method giving a partial solution of the considered problem. Our approach allows us to recover $$X_0(23)$$ and to construct another 18 distinct explicit curves of genus 2 defined over $$\mathbb Q$$ whose Jacobians have a rational point of order 11. Of these 19 curves, 10 do not have any rational Weierstrass point, and 9 have a rational Weierstrass point. None of these curves are $$\overline {\mathbb Q}$$-isomorphic to each other, nor $$\overline {\mathbb Q}$$-isomorphic to an element of Flynn’s family $$\mathfrak F_t$$. Finally, the Jacobians of these new curves are absolutely simple.

### MSC:

 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11G10 Abelian varieties of dimension $$> 1$$ 14G25 Global ground fields in algebraic geometry 14H40 Jacobians, Prym varieties

Zbl 0757.14025
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