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Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields

Published online by Cambridge University Press:  20 November 2018

Florian Breuer  and
Bo-Hae Im
Florian Breuer
Affiliation:
Department of Mathematical Sciences, University of Stellenbosch, Stellenbosch 7600, South Africa e-mail:, fbreuer@sun.ac.za
Bo-Hae Im
Affiliation:
Department of Mathematics, Chung-Ang University, 221 Haukseok-dong, Dongjak-gu, Seoul 156-756, South Korea e-mail:, imbh@cau.ac.kr
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Abstract

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Let $k$ be a global field, $\bar{k}$ a separable closure of $k$, and ${{G}_{k}}$ the absolute Galois group Gal$(\bar{k}/k)$ of $\bar{k}$ over $k$. For every $\sigma \,\in \,{{G}_{K}}$, let ${{\bar{k}}^{\sigma }}$ be the fixed subfield of $\bar{k}$ under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known that the Mordell–Weil group $E({{\bar{k}}^{\sigma }})$ has infinite rank. We present a new proof of this fact in the following two cases. First, when $k$ is a global function field of odd characteristic and $E$ is parametrized by a Drinfeld modular curve, and secondly when $k$ is a totally real number field and $E/k$ is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on $E$ defined over ring class fields.

Keywords

11G05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 3 , 01 June 2008 , pp. 481 - 490
Copyright
Copyright © Canadian Mathematical Society 2008

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