Heegner Points over Towers of Kummer Extensions

Darmon, Henri ; Tian, Ye

Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 1060-1081

Résumé

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.

Détail

DOI : 10.4153/CJM-2010-039-8

Mots-clés : 11G05, 11R23, 11F46

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