Tohoku Mathematical Journal

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

Ki-ichiro Hashimoto

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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.

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

First available in Project Euclid: 3 May 2007

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Zentralblatt MATH identifier

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]


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.

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  • [Brl] A. BRUMER, The rank of J0(N), Asterisque 228 (1995), 41-68.
  • [Br2] A. BRUMER, Curves with real multiplication, inpreparation
  • [Br3] A. BRUMER, Exercises diedraux et courbes a multiplication reelles, Actes du Seminaire de theorienombre de Paris (1989/1990), Birkhauser, Boston, inpreparation.
  • [CF] J. W. S. CASSELS AND E. V. FLYNN, Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, London Math. Soc. Lecture Note Ser. 230, Cambridge Univ. Press, 1996.
  • [Hg] Y. HASEGAWA, (g-curves over quadratic fields, Manuscripta Math.94 (1997), 347-364
  • [HHM] Y. HASEGAWA, K. HASHIMOTO AND F. MOMOSE, g-curve a n (j QM-curves, International J. Math. 10-7 (1999), 1011-1036.
  • [Ha] K. HASHIMOTO, -curves of degree 5 and abelian surfaces of GL2-type, Manuscripta Math. 98 (1999), 165-182.
  • [HM] K. HASHIMOTO AND N. MURABAYASHI, Shimura curves as intersections of Humbert surfaces and defin ing equations of QM-curves of genus two, Tohoku Math.J. 47 (1995), 271-296.
  • [Hum] G. HUMBERT, Sur les functions abeliennes singulieres, (Euvres de G. Humbert 2, pub. par les soins d Pierre Humbert et de Gaston Julia, Paris, Gauthier-Villars (1936), 297-401.
  • [GH] P. GRIFFITH AND J. HARRIS, On Cayley's explicit solution to Poncelet's porism, Enseigne. Math.II, Ser 24(1978), 31-40.
  • [Ko] T. KONDO, On certain family of sexics and their Galois group (in Japanese), 165-175, Proceedings of th 12-th Symposium on Algebraic combinatorics, 1995.
  • [Mae] T. MAEDA, Noether's problem for 5, J. Algebra 125 (1989), 418-430
  • [Mes.l] F. MESTRE, Courbes hyperelliptiques a multiplications reelles, C. R. Acad. Sci. Paris Ser. I Math. 30 (1988), 721-724.
  • [Mes2] F. MESTRE, Families de courbes hyperelliptiques a multiplications reelles, Arithmetic Algebraic Geome try, 193-208, Birkhauser Boston, Boston, MA, 1991.
  • [Se] J. -P. SERRE, Topics in Galois Theory, Research Notes in Mathematics 1, Jones and Bartlett Publ., Boston, MA, 1992.
  • [Sh] G. SHIMURA, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc.Japan11, Princeton University Press, Princeton, N. J., 1971.