Drinfeld modules and A-motives s are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and [Formula: see text]-motives, one can construct their [Formula: see text]-adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for [Formula: see text]-motives. Here, we clarify the rank-one case for A-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.

中文翻译：

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A-动机的 v-adic Galois 表示的非开放性

Drinfeld 模和 A-motives 是椭圆曲线和阿贝尔簇的函数场类似物。对于 Drinfeld 模块和 [公式：见文本]-动机，可以构建它们的 [公式：见文本]-adic 伽罗瓦表示并询问图像是否是开放的。对于 Drinfeld 模块，Richard Pink 和他的合著者已经回答了这个问题；然而，对于[公式：见文本]-动机，这个问题尚未得到解决。在这里，我们阐明了 A 动机的秩一情况，并表明当且仅当虚拟维度与地面场的特征素数时，伽罗瓦的图像是开放的。