2023/10/31 by Liang, Pei-Xin, Oki, Yasuhiro, Yu, Chia-Fu
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2310.20133
In this paper we investigate the Tate--Shafarevich group Sha1(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 Sha1(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 Sha1(k, T), and of the main result of Pollio [Pure App. Math. Q., 2014] for the case where the étale k-algebra in question is a product of two abelian extensions. Moreover, we improve the explicit description of Sha1(k, T) in Lee [J. Pure Appl. Alg., 2022] by removing an intersection condition.