Skip to main content
SpringerLink
Log in

Stable Models of Elliptic Curves, Ring Class Fields, and Complex Multiplication

  • Conference paper
Algorithmic Number Theory (ANTS 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3076))

Included in the following conference series:

  • 971 Accesses

Abstract

We introduce a new model for elliptic curves over rings of odd characteristic, and study its properties and its utility in numerical computations. They turn to be particularly interesting for elliptic curves with complex multiplication, for which they provide very simple stable equations. The invariants associated to these models allow an easy construction of ring class fields of certain imaginary quadratic orders, with interesting theoretical consequences and practical utility in numerical computations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atkin, O., Morain, F.: ‘Elliptic curves and primality proving’. Math. Comp. 61, 29–68 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berwick, W.E.H.: Modular invariants expressible in terms of quadratic and cubic irrationalities. Proc. London Math. Soc. 28, 53–69 (1928)

    Article  Google Scholar 

  3. Cohen, H.: Advanced Topics in Computational Number Theory. Graduate Texts in Math., vol. 193. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  4. Cox, D.: Primes of the form x2 + ny2: Fermat. In: Class Field Theory and Complex Multiplication, John Wiley & Sons, Chichester (1989)

    Google Scholar 

  5. Enge, A., Morain, F.: Comparing Invariants for Class Fields of Imaginary Quadratic Fields. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 252–266. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Gee, A., Stevenhagen, P.: Generating class invariant using Shimura reciprocity. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 441–453. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  8. Kaltofen, E., Yui, N.: Explicit construction of the Hilbert class fields of imaginary quadratic fields by integer lattice reduction. In: Number Theory, New York Seminars 1989-1990, pp. 149–202. Springer, Heidelberg (1991)

    Google Scholar 

  9. McKean, H., Moll, V.: Elliptic curves: function theory, geometry, arithmetic. Cambridge Univ. Press, Cambridge (1999)

    MATH  Google Scholar 

  10. Schertz, R.: Die singulären Werte der Weberschen Funktionen f, f1, f2, γ2, γ3’. J. Reine Angew. Math. 287, 46–74 (1976)

    Article  MathSciNet  Google Scholar 

  11. Schertz, R.: Weber’s class invariants revisited. J. Théorie des Nombres de Bordeaux (14), 325–343 (2002)

    Google Scholar 

  12. Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan, 11, Princeton University Press (1971)

    Google Scholar 

  13. Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Math., vol. 151. Springer, Heidelberg (1994)

    MATH  Google Scholar 

  14. Yui, N., Zagier, D.: On the singular values of Weber modular functions. Math. of Computation 66(220), 1645–1662 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament de Matemàtica Aplicada IV., Escola Universitària Politècnica de Vilanova i la Geltrú, Avinguda Víctor Balaguer s/n, E-08800, Vilanova i la Geltrú

    Jordi Guàrdia & Eugenia Torres

  2. Departament de Matemàtica Aplicada II., Facultat de Matemàtiques i Estadística, c/ Pau Gargallo, 5, E-08028, Barcelona

    Montserrat Vela

Authors
  1. Jordi Guàrdia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eugenia Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Montserrat Vela
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, University of South Carolina , SC 29208, Columbia, USA

    Duncan Buell

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Guàrdia, J., Torres, E., Vela, M. (2004). Stable Models of Elliptic Curves, Ring Class Fields, and Complex Multiplication. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-24847-7_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22156-2

  • Online ISBN: 978-3-540-24847-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation