Jacobians of Genus-2 Curves with a Rational Point of Order 11

Abstract

On the one hand, it is well known that Jacobians of (hyper)elliptic curves defined over $\Q$ 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 $\Q$. 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 Flynn found such a family $\Fl_t$ 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 $\Fl_t$. 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 $\Q$ 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 $\Qb$-isomorphic to each other, nor $\Qb$-isomorphic to an element of Flynn's family $\Fl_t$. Finally, the Jacobians of these new curves are absolutely simple.