Let $k$ be a global field, $overline{k}$ a separableclosure of $k$, and $G_k$ the absolute Galois group$Gal(overline{k}/k)$ of $overline{k}$ over $k$. For every$sigmain G_k$, let $ks$ be the fixed subfield of $overline{k}$under $sigma$. Let $E/k$ be an elliptic curve over $k$. It is knownthat the Mordell--Weil group $E(ks)$ has infinite rank. We present anew proof of this fact in the following two cases. First, when $k$is a global function field of odd characteristic and $E$ isparametrized by a Drinfeld modular curve, and secondly when $k$ is atotally real number field and $E/k$ is parametrized by a Shimuracurve. In both cases our approach uses the non-triviality of asequence of Heegner points on $E$ defined over ring class fields.
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机译：假设$ k $为全局字段，$ overline {k} $为$ k $的可分离闭包，$ G_k $为$ overline {k} $超过$$的绝对Galois组$ Gal（overline {k} / k）$ k $。对于G_k $中的每个$ sigma，令$ ks $为$ sigma $下$ overline {k} $的固定子字段。令$ E / k $为$ k $的椭圆曲线。众所周知，莫德尔-威尔集团$ E（ks）$具有无限排名。在以下两种情况下，我们提供了这一事实的新证明。首先，当$ k $是具有奇特征的全局函数字段，而$ E $由Drinfeld模数曲线参数化时，第二，当$ k $是完全实数字段且$ E / k $由Shimuracurve参数化时。在这两种情况下，我们的方法都使用在环类字段上定义的$ E $上Heegner点的无序性的平凡性。
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