## Tohoku Mathematical Journal

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

Ki-ichiro Hashimoto

#### 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

https://projecteuclid.org/euclid.tmj/1178207751

Digital Object Identifier
doi:10.2748/tmj/1178207751

Mathematical Reviews number (MathSciNet)
MR1793932

Zentralblatt MATH identifier
0993.11031

#### 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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