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Heegner Points over Towers of Kummer Extensions

Published online by Cambridge University Press:  20 November 2018

Henri Darmon
Affiliation:
Department of Mathematics, McGill University, Montréal, PQ H3A 2T5
Ye Tian
Affiliation:
Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing100190, P.R.China
Corresponding
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Abstract

Let $E$ be an elliptic curve, and let ${{L}_{n}}$ be the Kummer extension generated by a primitive ${{p}^{n}}$ -th root of unity and a ${{p}^{n}}$ -th root of $a$ for a fixed $a\,\in \,{{\mathbb{Q}}^{\times }}\,-\,\left\{ \pm 1 \right\}$ . A detailed case study by Coates, Fukaya, Kato and Sujatha and $V$ . Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of $E$ over ${{L}_{n}}$ in certain cases. The aim of this note is to explain how some of these predictions might be accounted for by Heegner points arising from a varying collection of Shimura curve parametrisations.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

References

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