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# Heegner Points over Towers of Kummer Extensions

Published:2010-05-20
Printed: Oct 2010
• Henri Darmon,
Department of Mathematics, McGill University, Montréal, PQ H3A2T5
• Ye Tian,
Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, P.R.China
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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 $a\in \mathbb{Q}^\times-\{\pm 1\}$. 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.
 MSC Classifications: 11G05 - Elliptic curves over global fields [See also 14H52] 11R23 - Iwasawa theory 11F46 - Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms