1. Cohomological properties of multinorm-one tori
- Author
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Liang, Pei-Xin, Oki, Yasuhiro, and Yu, Chia-Fu
- Subjects
Mathematics - Number Theory - Abstract
In this paper we investigate the Tate--Shafarevich group Sha^1(k, T) of a multinorm-one torus $T$ over a global field $k$. We establish a few functorial maps among cohomology groups and explore their relations. Using these properties and relations we obtain a few basic structural results for Sha^1(k, T) and extend a few results of Bayer-Fluckiger--Lee--Parimala [Adv. in Math., 2019] to some more general multinorm-one tori. We also give a uniform proof of a result of Demarche--Wei for a criterion of the vanishing of Sha^1(k, T), and of the main result of Pollio [Pure App. Math. Q., 2014] for the case where the \'etale $k$-algebra in question is a product of two abelian extensions. Moreover, we improve the explicit description of Sha^1(k, T) in Lee [J. Pure Appl. Alg., 2022] by removing an intersection condition., Comment: 26 pages, comments welcome
- Published
- 2023