| Reference : Jacobians of genus 2 curves with a rational point of order 11 |
| Scientific journals : Article | |||
| Physical, chemical, mathematical & earth Sciences : Mathematics | |||
| http://hdl.handle.net/10993/506 | |||
| Jacobians of genus 2 curves with a rational point of order 11 | |
| English | |
Leprévost, Franck  [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC) >] | |
| Bernard, Nicolas [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC) >] | |
| Pohst, Michael [> >] | |
| 2009 | |
| Experimental Mathematics | |
| A K Peters | |
| 18 | |
| 1 | |
| 65-70 | |
| Yes (verified by ORBilu) | |
| International | |
| 1058-6458 | |
| 1944-950X | |
| Natick | |
| MA | |
| [en] 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 non-trivial $l$-rank. On the other hand, it is also well-known that $11$ is the least prime number which 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 to partially solving 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 $\Q$ and 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. | |
| http://hdl.handle.net/10993/506 | |
| 10.1080/10586458.2009.10128884 |
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