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Proceedings of the American Mathematical Society
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
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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.


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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
PII: S 0002-9939(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