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Elliptic Curves over the Perfect Closure of a Function Field

Published online by Cambridge University Press:  20 November 2018

Dragos Ghioca*
Affiliation:
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4 e-mail: [email protected]
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Abstract

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We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Baker, M. H. and Silverman, J. H., A lower bound for the canonical height on abelian varieties over abelian extensions. Math. Res. Lett. 11(2004), no. 2–3, 377396.Google Scholar
[2] David, S. and Hindry, M., Minoration de la hauteur de Néron-Tate sur les variétés abéliennes de type C. M. J. Reine Angew. Math. 529(2000), 174.Google Scholar
[3] Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(1979), no. 4, 391401.Google Scholar
[4] Ghioca, D. and Moosa, R., Division points of subvarieties of isotrivial semi-abelian varieties. Int. Math. Res. Not. 2006, Art. ID 65437.Google Scholar
[5] Goldfeld, D. and Szpiro, L., Bounds for the order of the Tate-Shafarevich group. Compositio Math. 97(1995), no. 1–2, 7187 Google Scholar
[6] Hindry, M. and Silverman, J. H., The canonical height and integral points on elliptic curves. Invent. Math. 93(1988), no. 2, 419450. doi:10.1007/BF01394340Google Scholar
[7] Hindry, M. and Silverman, J. H., On Lehmer's conjecture for elliptic curves. In: Séminaire de Théorie des Nombres, Paris 1988-1989, Progr. Math. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 103116.Google Scholar
[8] Kim, M., Purely inseparable points on curves of higher genus. Math. Res. Lett. 4(1997), no. 5, 663666.Google Scholar
[9] Lang, S., Fundamentals of Diophantine geometry. Springer-Verlag, New York, 1983.Google Scholar
[10] Lang, S., Number theory. III. Diophantine geometry. In: Encyclopaedia of Mathematical Sciences 60, Springer-Verlag, Berlin, 1991.Google Scholar
[11] Laurent, M., Minoration de la hauteur de Néron-Tate. In: Séminaire de théorie des nombres, Paris 1981–82, Progr. Math. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 137151.Google Scholar
[12] Lehmer, D. H., Factorization of certain cyclotomic functions. Ann. of Math. (2) 34(1933), no. 3, 461479. doi:10.2307/1968172Google Scholar
[13] Levin, M., On the group of rational points on elliptic curves over function fields. Amer. J. Math. 90(1968), 456462. doi:10.2307/2373538Google Scholar
[14] Masser, D. W., Counting points of small height on elliptic curves. Bull. Soc. Math. France 117(1989), no. 2, 247265.Google Scholar
[15] Poonen, B., Local height functions and the Mordell-Weil theorem for Drinfeld modules. Compositio Math. 97(1995), no. 3, 349368.Google Scholar
[16] Scanlon, T., A positive characteristic Manin-Mumford theorem. Compositio Math. 141(2005), no. 6, 13511364. doi:10.1112/S0010437X05001879Google Scholar
[17] Serre, J.-P., Lectures on the Mordell-Weil theorem. Aspects of Mathematics E15, Friedr. Vieweg & Sohn, Braunschweig, 1989.Google Scholar
[18] Silverman, J. H., The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986.Google Scholar
[19] Silverman, J. H., Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics 151, Springer-Verlag, New York, 1994.Google Scholar
[20] Silverman, J. H., A lower bound for the canonical height on elliptic curves over abelian extensions. J. Number Theory 104(2004), no. 2, 353372. doi:10.1016/j.jnt.2003.07.001Google Scholar