Tohoku Mathematical Journal
2000
December
SECOND SERIES VOL. 52, NO. 4

Tohoku Math. J.
52 (2000), 475-488

Title ON BRUMER'S FAMILY OF RM-CURVES OF GENUS TWO

Author Ki-ichiro Hashimoto

(Received June 30, 1997, revised May 16, 2000)
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.

2000 Mathematics Subject Classification. Primary 11G30; Secondary 11G10, 14H10, 12F12.

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