Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T07:18:34.795Z Has data issue: false hasContentIssue false
  • English
  • Français

Mordell–Weil Groups and the Rank of Elliptic Curves over Large Fields

Published online by Cambridge University Press:  20 November 2018

Bo-Hae Im
Bo-Hae Im*
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112 e-mail: im@math.utah.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $K$ be a number field, $\bar{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\ne O$ and $3P\ne O$, then for each $\sigma \,\in \,\text{Gal(}\overline{K}/K\text{)}$, the Mordell–Weil group $E({{\overline{K}}^{\sigma }})$ of $E$ over the fixed subfield of $\bar{K}$ under $\sigma $ has infinite rank.

Keywords

11G05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 4 , 01 August 2006 , pp. 796 - 819
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Frey, G. and Jarden, M., Approximation theory and the rank of abelian varieties over large algebraic fields. Proc. London Math. Soc. 28(1974), 112–128.Google Scholar
[2] Fried, M. and Jarden, M., Field Arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete 11, Springer-Verlag, Berlin, 1986.Google Scholar
[3] Fried, M. and Jarden, M., Diophantine properties of subfields of . Amer. J. Math. 100(1978), no. 1, 653–666.Google Scholar
[4] Geyer, W.-D., Galois groups of intersections of local fields. Israel J. Math. 30(1978), no. 4, 382–396.Google Scholar
[5] Griffiths, P. and Harris, J., Principles of Algebraic Geometry. Wiley-Interscience Publication, New York, 1978.Google Scholar
[6] Hartshorne, R., Algebraic Geometry. Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.Google Scholar
[7] Im, B., Heegner points and Mordell–Weil groups of elliptic curves over large fields. Submitted for publication, 2003.Google Scholar
[8] Im, B. and Larsen, M., Weak approximation for linear systems of quadrics. To appear in J. Number Theory.Google Scholar
[9] Isaacs, I. M., Algebra: A Graduate Course. Brooks/Cole, Pacific Grove, CA, 1994.Google Scholar
[10] Lang, S., Algebra. Third edition. Addison-Wesley, 1993.Google Scholar
[11] Lang, S., Fundamentals of Diophantine Geometry. Springer-Verlag, New York, 1983.Google Scholar
[12] Larsen, M., Rank of elliptic curves over almost algebraically closed fields. Bull. London Math. Soc. 35(2003), no. 6, 817–820.Google Scholar
[13] Morandi, P., Field and Galois Theory. Graduate Texts in Mathematics 167, Springer-Verlag, New York, 1996.Google Scholar
[14] Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften 323, Springer, Berlin, 2000.Google Scholar
[15] Rudin, W., Real and complex analysis. Third edition. McGraw-Hill, New York, 1987.Google Scholar
[16] Serre, J. P., A Course in Arithmetic. Graduate Texts in Mathematics 7, Springer-Verlag, New York, 1973.Google Scholar
[17] Serre, J. P., Local Fields. Graduate Texts in Mathematics 67, Springer-Verlag, New York, 1979.Google Scholar
[18] Silverman, J. H., Integer points on curves of genus 1. J. London Math. Soc. 28(1983), no. 1, 1–7.Google Scholar
[19] Silverman, J. H., The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986.Google Scholar