### Abstract

We reconstruct Brumer’s family with 3-parameters of curves of genus two whose jacobian varieties admit a real multiplication of discriminant 5. Our method is based on the descent theory in geometric Galois theory which can be compared with a classical problem of Noether. Namely, we first construct a 3-parameter family of polynomials f(X) of degree 6 whose Galois group is isomorphic to the alternating group A_{5}. Then we study the family of curves defined by Y^{2} = f(X), showing that they are equivalent to Brumer's family. The real multiplication will be described in three distinct ways, i.e., by Humbert's modular equation, by Poncelet's pentagon, and by algebraic correspondences.

Original language | English |
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Pages (from-to) | 475-488 |

Number of pages | 14 |

Journal | Tohoku Mathematical Journal |

Volume | 52 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2000 Jan 1 |

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### ASJC Scopus subject areas

- Mathematics(all)