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.