Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The rank of elliptic curves with rational 2-torsion points over large fields


Author: Bo-Hae Im
Journal: Proc. Amer. Math. Soc. 134 (2006), 1623-1630
MSC (2000): Primary 11G05
Published electronically: December 15, 2005
MathSciNet review: 2204272
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a number field, $ \overline{K}$ an algebraic closure of $ K$, $ G_K$ the absolute Galois group $ \operatorname{Gal}(\overline{K}/K)$, $ K_{ab}$ the maximal abelian extension of $ K$ and $ E/K$ an elliptic curve defined over $ K$. In this paper, we prove that if all 2-torsion points of $ E/K$ are $ K$-rational, then for each $ \sigma\in G_K$, $ E((K_{ab})^{\sigma})$ has infinite rank, and hence $ E(\overline{K}^{\sigma})$ has infinite rank.


References [Enhancements On Off] (What's this?)

  • 1. B. Im: Mordell-Weil groups and the rank over large fields of elliptic curves over large fields, arXiv: math.NT/0411533, to appear in Canadian J. Math.
  • 2. B. Im: Heegner points and Mordell-Weil groups of elliptic curves over large fields, arXiv: math.NT/0411534, submitted for publication, 2003.
  • 3. Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605
  • 4. Michael Larsen, Rank of elliptic curves over almost separably closed fields, Bull. London Math. Soc. 35 (2003), no. 6, 817–820. MR 2000029, 10.1112/S0024609303002431
  • 5. Joseph H. Silverman, Integer points on curves of genus 1, J. London Math. Soc. (2) 28 (1983), no. 1, 1–7. MR 703458, 10.1112/jlms/s2-28.1.1

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11G05

Retrieve articles in all journals with MSC (2000): 11G05


Additional Information

Bo-Hae Im
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email: im@math.utah.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08494-7
Received by editor(s): January 28, 2005
Published electronically: December 15, 2005
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2005 American Mathematical Society