Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extensiongenerated by a primitive $p^n$-th root of unity and a $p^n$-th root of$a$ for a fixed $ain mathbb{Q}^ imes-{pm 1}$. A detailed case studyby Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led theseauthors to predict unbounded and strikingly regular growth for therank of $E$ over $L_n$ in certain cases. The aim of this note is toexplain how some of these predictions might be accounted for byHeegner points arising from a varying collection of Shimura curveparametrisations.
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机译：假设$ E $为椭圆曲线，而$ L_n $为Kummer扩展，它由原始的$ p ^ n $个单位根和一个固定的$ ain的$ p ^ n $个根生成mathbb {Q} ^ imes- {pm 1} $。 Coates，Fukaya，Kato和Sujatha和V.Dokchitser进行的详细案例研究使这些作者预测在某些情况下，$ E $超过$ L_n $的排名将无限制且惊人地有规律地增长。本说明的目的是说明如何通过对Shimura曲线参数集进行不同收集而产生的Heegner点来解释其中的一些预测。
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