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 ORBi^{lu}) | |

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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