### 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 |

Publication status | Published - 2000 Dec |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Tohoku Mathematical Journal*,

*52*(4), 475-488.

**On Brumer's family of RM-curves of genus two.** / Hashimoto, Kiichiro.

Research output: Contribution to journal › Article

*Tohoku Mathematical Journal*, vol. 52, no. 4, pp. 475-488.

}

TY - JOUR

T1 - On Brumer's family of RM-curves of genus two

AU - Hashimoto, Kiichiro

PY - 2000/12

Y1 - 2000/12

N2 - 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 A5. Then we study the family of curves defined by Y2 = 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.

AB - 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 A5. Then we study the family of curves defined by Y2 = 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.

UR - http://www.scopus.com/inward/record.url?scp=0034343835&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034343835&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034343835

VL - 52

SP - 475

EP - 488

JO - Tohoku Mathematical Journal

JF - Tohoku Mathematical Journal

SN - 0040-8735

IS - 4

ER -