Tohoku Mathematical Journal

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

Ki-ichiro Hashimoto

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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 $A_5$. 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.

Article information

Source
Tohoku Math. J. (2), Volume 52, Number 4 (2000), 475-488.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1178207751

Digital Object Identifier
doi:10.2748/tmj/1178207751

Mathematical Reviews number (MathSciNet)
MR1793932

Zentralblatt MATH identifier
0993.11031

Subjects
Primary: 14H25: Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
Secondary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25] 12F10: Separable extensions, Galois theory 14H40: Jacobians, Prym varieties [See also 32G20]

Citation

Hashimoto, Ki-ichiro. On Brumer's family of {RM}-curves of genus two. Tohoku Math. J. (2) 52 (2000), no. 4, 475--488. doi:10.2748/tmj/1178207751. https://projecteuclid.org/euclid.tmj/1178207751


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