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Computation of Sets of Rational Points of Genus-3 Curves via the Dem'Janenko–Manin Method

Published online by Cambridge University Press:  01 February 2010

Martine Girard  and
Leopoldo Kulesz
Martine Girard
Affiliation:
School of Mathematics and Statistics, University of Sydney NSW 2006, Australia, girard@maths.usyd.edu.au, http://www.maths.usyd.edu.au/u/girard/
Leopoldo Kulesz
Affiliation:
Instituto de Desarollo Humano, Universidad National de General Sarmiento, Roca 850 San Miguel, Pcia. de Buenos Aires. 1663, Argentina, lk@delzorzal.com.ar

Abstract

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The authors construct two families of genus-3 curves defined over Q(t) with three independent morphisms to an elliptic curve of rank at most two. They give explicit examples of an application of the Dem'janenko-Manin method that completely determines both the set of the Q(t)-rational points of the curves under consideration, and the set of Q-rational points of some specialisations.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 267 - 300
Copyright
Copyright © London Mathematical Society 2005

References

1.Auer, R. and Top, J., ‘Some genus 3 curves with many points’, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci. 2369 (Springer, Berlin, 2002)163–171.CrossRefGoogle Scholar
2.Bombieri, E., ‘The Mordell conjecture revisited’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990) 615–640.Google Scholar
3.Bosma, W. and Cannon, J.(eds), ‘Handbook of Magma functions’, http://magma.maths.usyd.edu.au/.Google Scholar
4.Bruin, N., ‘The diophantine equations x2±y4 = ±z6 and x2+ y8 = z3’, Compositio Math, 118 (1999) 305–321.CrossRefGoogle Scholar
5.Bruin, N. and Flynn, E.V., ‘n-covers of hyperelliptic curves’, Math. Proc. Cambridge Philos. Soc. 134 (2003) 397–405.CrossRefGoogle Scholar
6.Cassels, J.W.S., ‘On a theorem of Dem'janenko’, J. London Math. Soc. 43 (1968) 61–66.CrossRefGoogle Scholar
7.Chabauty, C., ‘Sur les points rationnels des variétés algébriques dont l'irrégularité est supérieure à la dimension’, C.R. Acad. Sci. Paris. 212 (1941) 1022–1024.Google Scholar
8.Coleman, R.F., ‘Effective Chabauty’, Duke Math. J. 52 (1985 )765–770.CrossRefGoogle Scholar
9.Cremona, J.E., ‘mwrank’ and other programs for elliptic curves over , http://www.maths.nott.ac.uk/personal/jec/ftp/progs/.Google Scholar
10.Cremona, J.E., Prickett, M. and Siksek, S., ‘Height differences bounds for elliptic curves over number fields’, J. Number Theory, to appear.Google Scholar
11.De Diego, T., ‘Points rationnels sur les families de courbes de genre au moins 2’, J. Number Theory 67 (1997) 85–114.CrossRefGoogle Scholar
12.Dem'Janenko, V.A., Rational points of a class of algebraic curves, Amer. Math. Soc. Transl., Ser. 1166 (Amer. Math. Soc, Providence, RI, 1968) 246–272.Google Scholar
13.Faltings, G. , ‘Endlichkeitssätze für abelsche Varietäten iiber Zahlkörpern’, Invent. Math.. 73 (1983) 349–366.CrossRefGoogle Scholar
14.Faltings, G.,‘Diophantine approximation on abelian varieties’, Ann. of Math.(2) 133 (1991) 549–576.CrossRefGoogle Scholar
15.Flynn, E.V., ‘A flexible method for applying Chabauty's theorem’, Compositio Math. 105 (1997) 79–94.CrossRefGoogle Scholar
16.Flynn, E.V., ‘Coverings of curves of genus 2’, Algorithmic number theory (Leiden, 2000), Lecture Notes in Comput. Sci. 1838 (Springer, Berlin, 2000) 65–84.CrossRefGoogle Scholar
17.Flynn, E.V., Poonen, B. and Schaefer, E.F., ‘Cycles of quadratic polynomials and rational points on a genus-2 curve’, Duke Math. J. 90 (1997) 435–463.CrossRefGoogle Scholar
18.Flynn, E.V. and Wetherell, J.L., ‘Covering collections and a challenge problem of Serre’, Acta Arith. 98 (2001) 197–205.CrossRefGoogle Scholar
19.Girard, M., ‘Points de weierstrass et courbes algébriques de genre 3’, Ph.D. thesis, Université Paris 7 (2000).Google Scholar
20.Girard, M. and Tzermias, P., ‘Group generated by the Weierstrass points of a plane quartic’, Proc. Amer. Math. Soc. 130 (2002) 667–672.CrossRefGoogle Scholar
21.Hindry, M. and Silverman, J.H., Diophantine geometry, Grad. Texts in Math. 201 (Springer, New York, 2000).CrossRefGoogle Scholar
22.Howe, E. W., Leprévost, F. and Poonen, B., ‘Large torsion subgroups of split Jacobians of curves of genus two or three’, Forum Math. 12 (2000) 315–364.CrossRefGoogle Scholar
23.Kulesz, L., ‘Application de la méthode de Dem'janenko-Manin à certaines families de courbes de genre 2 et 3’, J.Number Theory 76 (1999) 130–146.CrossRefGoogle Scholar
24.Kulesz, L., ‘Courbes elliptiques de rang ≥ 5 sur avec un groupe de torsion isomorphe à’, C.R.Acad. Sci. Paris Sér I Math. 329 (1999) 503–506.CrossRefGoogle Scholar
25.Manin, Ju.I., ‘Rational points on algebraic curves over function fields’, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963) 1395–1440.Google Scholar
26.Manin, Ju.I., ‘The p-torsion of elliptic curves is uniformly bounded’, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 459–465.Google Scholar
27.Rémond, G., ‘Inégalité de Vojta en dimension supérieure’, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4)29 (2000) 101–151.Google Scholar
28.Serre, J.-P., Lectures on the Mordell-Weil theorem, Aspects of Math.E 15 (Vieweg, 1989).CrossRefGoogle Scholar
29.Shioda, T., ‘On the Mordell-Weil lattices’, Comment. Math. Univ. St. Paul 39 (1990) 211–240.Google Scholar
30.Shioda, T., ‘On the rank of elliptic curves over Q(t)arising from K3 surfaces’, Comment. Math. Univ. St. Paul. 43 (1994) 117–120.Google Scholar
31.Siksek, S., ‘Infinite descent on elliptic curves’, Rocky Mountain J.Math. 25 (1995) 1501–1538.CrossRefGoogle Scholar
32.Silverman, J.H., ‘The arithmetic of elliptic curves, Grad.Texts in Math. 106(Springer, New York, 1986).CrossRefGoogle Scholar
33.Silverman, J.H., ‘Rational points on certain families of curves of genus at least 2’, Proc. London Math. Soc.(3)55 (1987) 465–481.CrossRefGoogle Scholar
34.Silverman, J.H., The difference between the Weil height and the canonical height on elliptic curves’, Math. Comp. 55 (1990) 723–743.CrossRefGoogle Scholar
35.Silverman, J.H., Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math. 151 (Springer, New York, 1994).CrossRefGoogle Scholar
36.Top, J., Hecke 1-series related with algebraic cycles or with siegel modular forms’, PhD thesis, Utrecht University(1989).Google Scholar
37. University of Sydney Computational Algebra Group, ‘Magma computational algebra system’, http://magma.maths.usyd.edu.au/.Google Scholar
38.Vojta, P., ‘Siegel's theorem in the compact case’, Ann. of Math. (2) 133 (1991) 509–548.CrossRefGoogle Scholar
39.Weil, A., ‘Remarques sur un mémoire d'Hermite’, Arch. Math. 5 (1954) 197–202.CrossRefGoogle Scholar
40.Wetherell, J.L., ‘Bounding the number of rational points on certain curves of high rank’, Ph.D.thesis, University of California, Berkeley (1997).Google Scholar