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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Heegner points and Mordell-Weil groups of elliptic curves over large fields

Author: Bo-Hae Im
Journal: Trans. Amer. Math. Soc. 359 (2007), 6143-6154
MSC (2000): Primary 11G05
Published electronically: June 4, 2007
MathSciNet review: 2336320
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Abstract: Let $ E/\mathbb{Q}$ be an elliptic curve defined over $ \mathbb{Q}$ of conductor $ N$ and let $ \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of an algebraic closure $ \overline{\mathbb{Q}}$ of $ \mathbb{Q}$. For an automorphism $ \sigma\in \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we let $ \overline{\mathbb{Q}}^{\sigma}$ be the fixed subfield of $ \overline{\mathbb{Q}}$ under $ \sigma$. We prove that for every $ \sigma\in \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the Mordell-Weil group of $ E$ over the maximal Galois extension of $ \mathbb{Q}$ contained in $ \overline{\mathbb{Q}}^{\sigma}$ has infinite rank, so the rank of $ E(\overline{\mathbb{Q}}^{\sigma})$ is infinite. Our approach uses the modularity of $ E/\mathbb{Q}$ and a collection of algebraic points on $ E$ - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer $ r$ and for a prime $ p$ prime to $ rN$, the rank of $ E$ over all the ring class fields of a conductor of the form $ rp^n$ is unbounded, as $ n$ goes to infinity.

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Bo-Hae Im
Affiliation: Department of Mathematics, Chung-Ang University, 221 Heukseok-dong, Dongjak-gu, Seoul 156-756, South Korea

Received by editor(s): August 4, 2004
Received by editor(s) in revised form: April 25, 2006
Published electronically: June 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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