Heegner points over towers of Kummer extensions

Abstract

Let E be an elliptic curve, and let L n be the Kummer extension generated by a primitive p nth root of unity and a p"-th root of a for a fixed a ε ℚ x -{±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. © Canadian Mathematical Society 2010.