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Galois criterion for torsion points of Drinfeld modules
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2021-07-06 , DOI: 10.1142/s1793042121500901 Chien-Hua Chen 1
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2021-07-06 , DOI: 10.1142/s1793042121500901 Chien-Hua Chen 1
Affiliation
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 𝔽 q [ T ] , the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over 𝔽 q ( T ) contains a rational 𝔩 -torsion point if the reduction of ϕ at almost all primes of 𝔽 q [ 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:
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𝔽 q [ 吨 ] ,这个问题本质上是问,直到同源,一个 Drinfeld 模块φ 超过𝔽 q ( 吨 ) 包含一个理性的𝔩 -扭力点,如果减少φ 在几乎所有素数𝔽 q [ 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3 Drinfeld 模块我们对答案是肯定的情况进行分类。
更新日期:2021-07-06
中文翻译:
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