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

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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
DOIs
Publication statusPublished - 2000 Jan 1

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ASJC Scopus subject areas

  • Mathematics(all)

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