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Minimal Models for 2-coverings of Elliptic Curves

Published online by Cambridge University Press:  01 February 2010

Michael Stoll
Affiliation:
School of Engineering and Science, International University Bremen, P.O.Box 750561, 28725 Bremen, Germany, m.stoll@iu-bremen.de, http://www.iu-bremen.de/directory/faculty/29179/
John E. Cremona
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, john.cremona@nottingham.ac.uk, http://www.maths.nottingham.ac.uk/personal/jec/

Abstract

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This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y2 = Q(x), where Q(x) has degree 4. These models are used in the method of 2-descent for computing the rank of an elliptic curve. The results described here are complete for unramified extensions of Q2 and Q3, and for all p-adic fields for p greater than or equal to 5. The primary motivation for this work was to complete the results of Birch and Swinnerton-Dyer, which are incomplete in the case of Q2. The results in this case (when applied to 2-coverings of elliptic curves over Q) yield substantial improvements in the running times of the 2-descent algorithm implemented in the program mwrank. The paper ends with a section on implementation and examples, and an appendix gives constructive proofs in sufficient detail to be used for implementation.

Type
Research Article
Copyright
Copyright © London Mathematical Society 2002

References

1Brumer, A., Kramer, K., ‘The rank of elliptic curves’, Duke Math. J. 44 (1977) 715–743.Google Scholar
2Birch, B. J., Swinnerton-Dyer, H.P.F., ‘Notes on elliptic curves, I’, J.Reine Angew. Math. 212 (1963) 7–25.Google Scholar
3Cremona, J. E., Algorithms for modular elliptic curves, 2nd edn (Cambridge University Press, 1997).Google Scholar
4Cremona, J. E., ‘Classical invariants and 2-descent on elliptic curves’, J.Symbolic Comput. 31 (2001) 71–87.Google Scholar
5Cremona, J. E., ‘mwrank’ and other programs for elliptic curves over ℚ, http://www.maths.nott.ac.uk/personal/jec/ftp/progs.Google Scholar
6Cremona, J. E., Serf, P., ‘Computing the rank of elliptic curves over real quadratic fields of class number 1’, Math. Comp. 68 (1999) 1187–1200.Google Scholar
7Silverman, J. H.:The arithmetic of elliptic curves, Grad. Texts in Math. 106 (Springer, 1986).Google Scholar
8Serf, P.: ‘The rank of elliptic curves over real quadratic number fields of class number 1’, Thesis, Universität des Saarlandes, 1995.Google Scholar
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