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Explicit Models of Genus 2 Curves with Split CM

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Algorithmic Number Theory (ANTS 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1838))

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Abstract

We outline a general algorithm for computing an explicit model over a number field of any curve of genus 2 whose (unpolarized) Jacobian is isomorphic to the product of two elliptic curves with CM by the same order in an imaginary quadratic field. We give the details and some examples for the case where the order has prime discriminant and class number one.

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References

  1. Eichler, M.: Zur Zahlentheorie der Quaternion-Algebren. J. Reine Angew. Math. 195, 127–151 (1955)

    Article  MathSciNet  Google Scholar 

  2. Gross, B., Zagier, D.: On singular moduli. J. Reine Angew. Math. 355, 191–220 (1985)

    MATH  MathSciNet  Google Scholar 

  3. Hayashida, T., Nishi, M.: On certain type of Jacobian varieties of dimension 2. Natur. Sci. Rep. Ochanomizu Univ. 16, 49–57 (1965)

    MATH  MathSciNet  Google Scholar 

  4. Hermite, C.: Oeuvres de Charles Hermite. Gauthiers-Villars, Paris (1905)

    MATH  Google Scholar 

  5. Igusa, J.: Arithmetic variety of moduli for genus two. Ann. of Math. 72, 612–649 (1960)

    Article  MathSciNet  Google Scholar 

  6. Mestre, J.-F.: Construction de courbes de genre 2 à partir de leurs modules. In: Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., vol. 94, pp. 313–334. Birkhäuser, Basel (1991)

    Google Scholar 

  7. Mumford, D.: Tata lectures on Theta I. Birkhauser. Birkhäuser, Basel (1983)

    Google Scholar 

  8. Narasimhan, M.S., Nori, M.V.: Polarisations on an abelian variety. Proc. Indian Acad. Sci. Math. Sci. 90, 125–128 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. Shimura, G.: On the field of rationality for an abelian variety. Nagoya Math. J. 45, 167–178 (1972)

    MATH  MathSciNet  Google Scholar 

  10. van Wamelen, P.: Examples of genus two CM curves defined over the rationals. Math. Comp. 68, 307–320 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Texas at Austin, TX, 78712, USA

    Fernando Rodriguez-Villegas

Authors
  1. Fernando Rodriguez-Villegas
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Editor information

Editors and Affiliations

  1. Mathematisch Instituut, Universiteit Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    Wieb Bosma

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© 2000 Springer-Verlag Berlin Heidelberg

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Rodriguez-Villegas, F. (2000). Explicit Models of Genus 2 Curves with Split CM. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_33

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  • DOI: https://doi.org/10.1007/10722028_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67695-9

  • Online ISBN: 978-3-540-44994-2

  • eBook Packages: Springer Book Archive

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