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Counting models of genus one curves

Published online by Cambridge University Press:  12 January 2011

MOHAMMAD SADEK
MOHAMMAD SADEK*
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB. e-mail: [email protected]
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Abstract

Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a (generalised) binary quartic when n = 2, a ternary cubic when n = 3 and a pair of quaternary quadrics when n = 4. In general, minimal genus one equations of degree n are not unique up to isomorphism. We explain how the number of these equations varies according to the Kodaira symbol of the Jacobian of C. Then we count these equations up to isomorphism over a number field of class number 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 3 , May 2011 , pp. 399 - 417
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]An, S. Y., Kim, S. Y., Marshall, D. C., Marshall, S. H., McCallum, W. G. and Perlis, A. R.Jacobians of genus one curves. J. Number Theory 90 (2) (2001), 304–315.CrossRefGoogle Scholar
[2]Birch, B. J. and Swinnerton-Dyer, H. P. F.Notes on elliptic curves I. J. Reine Angew. Math. 212 (1963), 7–25.CrossRefGoogle Scholar
[3]Bosma, W., Cannon, J. and Playoust, C.The Magma algebra system I: The user language. J. Symb. Comb. 24 (1997), 235–265.CrossRefGoogle Scholar
[4]Bromwich, T.Quadratic forms and their classification by means of invariant factors. Cambridge Tracts in Mathematics and Mathematical Physics (1906).Google Scholar
[5]Cremona, J. E., Fisher, T. A. and Stoll, M. Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, preprint.CrossRefGoogle Scholar
[6]Fisher, T. A.A new approach to minimising binary quartics and ternary cubics. Math. Res. Lett. 14 (2007), 597–613.CrossRefGoogle Scholar
[7]Fisher, T. A.The invariants of a genus one curve. Proc. Lond. Math. Soc. 97 (3) (2008), 753–782.CrossRefGoogle Scholar
[8]Liu, Q.Modèles entiers des courbes hyperelliptiques sur un corps de valuations discrète. Trans. Amer. Math. Soc. 348 (11) (November 1996), 4577–4610.CrossRefGoogle Scholar
[9]Liu, Q.Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics vol. 6 (Oxford University Press, 2002).CrossRefGoogle Scholar
[10]Lorenzini, D. Models of curves and wild ramification, preprint.Google Scholar
[11]Poonen, B.An explicit algebraic family of genus-one curves violating the Hasse principle. J. Théor. Nombres Bordeaux 13 (1) (2001), 263–274.CrossRefGoogle Scholar
[12]Sadek, M. Minimal genus one curves, preprint.Google Scholar
[13]Sadek, M. Models of genus one curves. PhD thesis. Cambridge University (2009).Google Scholar
[14]Silverman, J.Advanced topics in the arithmetic of elliptic curves. GTM 151. (Springer-Verlag, 1995).Google Scholar
[15]Stoll, M. and Cremona, J. E.Minimal models for 2-coverings of elliptic curves. LMS J. Comput. Math. 5 (2002) 220–243.CrossRefGoogle Scholar
[16]Womack, T. O. Explicit descent on elliptic curves. PhD thesis. University of Nottingham (2003).Google Scholar