Computing Tate–Shafarevich Groups of Multinorm One Tori of Kummer Type

A multinorm one torus associated to a commutative étale algebra L over a global field k is of Kummer type if each factor of L is a cyclic Kummer extension. In this paper we compute the Tate–Shafarevich group of such tori based on general formulas of Lee [4]. Our aim is to illustrate various invariants in Lee's formulas by this class of tori, especially from the computational aspect. We also implement an effective algorithm using SageMath which computes the Tate–Shafarevich groups when each factor of L is contained in a fixed concrete bicyclic extension of a cyclotomic field k.