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On the Jacobian of the Klein curve


Author: Despina T. Prapavessi
Journal: Proc. Amer. Math. Soc. 122 (1994), 971-978
MSC: Primary 14H40; Secondary 14H45, 14H52
DOI: https://doi.org/10.1090/S0002-9939-1994-1212286-1
MathSciNet review: 1212286
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Abstract: Its is known that the Jacobian J of the Klein curve is isogenous to $ {{\mathbf{E}}^3}$ for a certain elliptic curve E. We compute explicit equations for E and prove that J is in fact isomorphic to $ {{\mathbf{E}}^3}$. We also identify the subgroup of J generated by the image of the Weierstrass points of the curve under an Albanese embedding, and we show that it is isomorphic to $ {\mathbf{Z}}/2{\mathbf{Z}} \times {({\mathbf{Z}}/7{\mathbf{Z}})^3}$.


References [Enhancements On Off] (What's this?)

  • [1] R. Coleman, Torsion points on curves and p-adic abelian integrals, Ann. of Math. (2) 121 (1985), 111-165. MR 782557 (86j:14014)
  • [2] -, Torsion points on Abelian etale coverings of $ {P^1} - \{ 0,1,\infty \} $, Trans. Amer. Math. Soc. 311 (1989), 185-208. MR 974774 (90a:11064)
  • [3] R. Greenberg, On the Jacobian variety of some algebraic curves, Compositio Math. 42 (1981), 345-359. MR 607375 (82j:14036)
  • [4] B. Gross and D. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978), 201-224. MR 0491708 (58:10911)
  • [5] F. Klein, Uber die Transformation siebenter Ordhang der elliptischen Functionen, Gesammelte Math. Abhandlungen III, vol. 84, Springer, Berlin, 1923.
  • [6] N. Koblitz and D. Rohrlich, Simple factors in the Jacobian of the Fermat curve, Canad. J. Math. 20 (1978), 1183-1205. MR 511556 (80d:14022)
  • [7] S. Lang, Complex multiplication, Springer-Verlag, New York, 1983. MR 713612 (85f:11042)
  • [8] J. Rotman, An introduction to homological algebra, Academic Press, New York, 1979. MR 538169 (80k:18001)
  • [9] D. Springer, Introduction to Riemann surfaces, Addison Wesley, Reading, MA, 1957. MR 0092855 (19:1169g)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1212286-1
Article copyright: © Copyright 1994 American Mathematical Society

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