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Continued fractionsin $ 2$-stage Euclidean quadratic fields

Authors: Xavier Guitart and Marc Masdeu
Journal: Math. Comp. 82 (2013), 1223-1233
MSC (2010): Primary 13F07, 11A55
Published electronically: October 15, 2012
MathSciNet review: 3008856
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Abstract: We discuss continued fractions on real quadratic number fields of class number $ 1$. If the field has the property of being $ 2$-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions. Although it is conjectured that all real quadratic fields of class number $ 1$ are $ 2$-stage euclidean, this property has been proven for only a few of them. The main result of this paper is an algorithm that, given a real quadratic field of class number $ 1$, verifies this conjecture, and produces as byproduct enough data to efficiently compute continued fraction expansions. If the field was not $ 2$-stage euclidean, then the algorithm would not terminate. As an application, we enlarge the list of known $ 2$-stage euclidean fields, by proving that all real quadratic fields of class number $ 1$ and discriminant less than $ 8000$ are $ 2$-stage euclidean.

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  • 1. George E. Cooke, A weakening of the Euclidean property for integral domains and applications to algebraic number theory. I, J. Reine Angew. Math. 282 (1976), 133–156. MR 0406973
  • 2. George E. Cooke, A weakening of the Euclidean property for integral domains and applications to algebraic number theory. II, J. Reine Angew. Math. 283/284 (1976), 71–85. MR 0406974
  • 3. George Cooke and Peter J. Weinberger, On the construction of division chains in algebraic number rings, with applications to 𝑆𝐿₂, Comm. Algebra 3 (1975), 481–524. MR 0387251
  • 4. J. E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Math. 51 (1984), no. 3, 275–324. MR 743014
  • 5. Henri Darmon and Adam Logan, Periods of Hilbert modular forms and rational points on elliptic curves, Int. Math. Res. Not. 40 (2003), 2153–2180. MR 1997296, 10.1155/S1073792803131108
  • 6. H. Davenport, Indefinite binary quadratic forms, and Euclid’s algorithm in real quadratic fields, Proc. London Math. Soc. (2) 53 (1951), 65–82. MR 0041883
  • 7. Lassina Dembélé, An algorithm for modular elliptic curves over real quadratic fields, Experiment. Math. 17 (2008), no. 4, 427–438. MR 2484426
  • 8. L. Dembélé, J. Voight, Explicit methods for Hilbert modular forms. To appear in ``Elliptic Curves, Hilbert modular forms and Galois deformations''. arXiv:1010.5727v2.
  • 9. Veikko Ennola, On the first inhomogeneous minimum of indefinite binary quadratic forms and Euclid’s algorithm in real quadratic fields, Ann. Univ. Turku. Ser. AI 28 (1958), 58. MR 0097356
  • 10. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
  • 11. Franz Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exposition. Math. 13 (1995), no. 5, 385–416. MR 1362867

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Additional Information

Xavier Guitart
Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain

Marc Masdeu
Affiliation: Department of Mathematics, Columbia University, New York, New York 10001

Received by editor(s): June 4, 2011
Received by editor(s) in revised form: September 6, 2011
Published electronically: October 15, 2012
Additional Notes: This work was partially supported by Grants MTM2009-13060-C02-01 and 2009 SGR 1220.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.