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Problèmes D'effectivité sur les Quartiques de Fermat

Published online by Cambridge University Press:  17 April 2009

Élie Cali  and
Alain Kraus
Élie Cali
Affiliation:
App. 231, 9 rue de Sèvres, 92100 Boulogne, France, e-mail: [email protected]
Alain Kraus
Affiliation:
Université Pierre et Marie Curie - Paris 6, Institut de Mathématiques, UMR 7586 du CNRS, 175 rue du Chevaleret, 75013 Paris, France, e-mail: [email protected]
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Let K be a number field. An element b ∈ K* being given, let Cb be the curve defined over K by the equation x4 + y4 = bz4. Let Cb(K) be the set of the K-rational points of Cb. This paper uses Dem'janenko and Manin type methods to obtain effective criteria for Cb(K) to be empty.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 1 , February 2007 , pp. 91 - 119
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Cremona, J.E., Prickett, M. et Siksek, S., ‘Height difference bounds for elliptic curves over number fields’, J. Number Theory 116 (2006), 42–68.CrossRefGoogle Scholar
[2]Dem'janenko, V.A., ‘The Indeterminate Equations x 6 + y 6 = az 2, x 6 + y 6 = az 3, x 4 + y 4 = az 4’, Amer. Math. Soc. Transl. 119 (1983), 27–34.Google Scholar
[3]Grigorov, G. et Rizov, J., ‘Heights on elliptic curves and the diophantine equation x 4 + y 4 = cz 4’, (preprint), Sophia University (1998).Google Scholar
[4]Kraus, A., ‘Quelques remarques à propos des invariants c 4, c 6 et ▵ d'une courbe elliptique’, Acta Arith. 54 (1989), 75–80.CrossRefGoogle Scholar
[5]Siksek, S., ‘Infinite descent on elliptic curves’, Rocky Mountain J. Math. 25 (1995), 1501–1538.CrossRefGoogle Scholar
[6]Silverman, J.H., ‘Lower bounds for height functions’, Duke Math. J. 51 (1984), 395–403.CrossRefGoogle Scholar
[7]Silverman, J.H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer-Verlag, New York, 1986).CrossRefGoogle Scholar
[8]Silverman, J.H., Rational points on certain families of curves of genus at least 2, Proc. London Math. Soc. 55 (1987), 465–481.CrossRefGoogle Scholar
[9]Simon, D., ‘Programme de calcul du rang des courbes elliptiques dans les corps de nombres, disponible à l'adresse’, http://www.math.unicaen.fr/-simon/.Google Scholar
[10]Tate, J., ‘Algorithm for determining the type of a singular fiber in an elliptic pencil, dans Modular Functions of One Variable IV’, Lecture Notes in Math. 476 (1975), 33–52.CrossRefGoogle Scholar