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

    Research output: Contribution to journalArticle

    11 Citations (Scopus)

    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 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.

    Original languageEnglish
    Pages (from-to)475-488
    Number of pages14
    JournalTohoku Mathematical Journal
    Volume52
    Issue number4
    Publication statusPublished - 2000 Dec

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    Genus
    Curve
    Multiplication
    Jacobian Varieties
    Modular Equations
    Galois Theory
    Pentagon
    Alternating group
    Noether
    Galois group
    Descent
    Discriminant
    Correspondence
    Isomorphic
    Distinct
    Polynomial
    Family

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

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

    In: Tohoku Mathematical Journal, Vol. 52, No. 4, 12.2000, p. 475-488.

    Research output: Contribution to journalArticle

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