In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal 𝔩 of 𝔽q[T], the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over 𝔽q(T) contains a rational 𝔩-torsion point if the reduction of ϕ at almost all primes of 𝔽q[T] contains a rational 𝔩-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

中文翻译：

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Drinfeld 模块扭转点的伽罗瓦准则

在本文中，我们制定了由 Lang 提出并由 Katz 研究的关于在数域上存在有理点的问题的 Drinfeld 模块模拟。给定一个最大理想𝔩的𝔽q[吨]，这个问题本质上是问，直到同源，一个 Drinfeld 模块φ超过𝔽q(吨)包含一个理性的𝔩-扭力点，如果减少φ在几乎所有素数𝔽q[吨]包含一个理性的𝔩-扭力点。与阿贝尔变种的情况类似，如果 Drinfeld 模块的秩为2, 但如果等级是负数3. 此外，对于排名3Drinfeld 模块我们对答案是肯定的情况进行分类。