AI 中文总结
该研究建立了伽罗瓦表示及其$p$进族形变相关精细Selmer群的垂直控制定理，为Coates–Sujatha的两个猜想提供新证据，还得到多类对象对应精细Selmer群的伪零性。
AI 中文摘要
本文建立了与伽罗瓦表示及其$p$进族形变相关的精细Selmer群的垂直控制定理。应用这些定理，我们为Coates–Sujatha的猜想A和猜想B提供了新证据，并得到了分圆特征、椭圆曲线、经典尖点新形式以及$p$-常尖点新形式的Hida族对应的精细Selmer群的伪零性。
英文摘要
In this article, we develop vertical control theorems for fine Selmer groups associated with Galois representations and their deformations in $p$-adic families. We apply them to provide new evidence for Conjectures A and B of Coates--Sujatha and to obtain pseudonullity of the fine Selmer groups for cyclotomic characters, elliptic curves, classical cuspidal newforms, and Hida families of $p$-ordinary cuspidal newforms.
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