Computing Tate-Shafarevich groups of multinorm one tori of Kummer type

TL;DR

This paper computes the Tate-Shafarevich groups of multinorm one tori of Kummer type over global fields, providing theoretical formulas and an effective SAGE algorithm for specific cases.

Contribution

It offers explicit computation methods for Tate-Shafarevich groups of Kummer-type multinorm tori, extending previous theoretical work with practical algorithms.

Findings

Explicit formulas for Tate-Shafarevich groups of Kummer-type tori

An effective SAGE algorithm for specific multinorm cases

Implementation results demonstrating the algorithm's utility

Abstract

A multinorm one torus associated to a commutative \'etale 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 recent works of Bayer-Fluckiger, T.-Y. Lee and Parimala, and of T.-Y.~Lee. We also implement an effective algorithm using SAGE which computes the Tate-Shafarevich groups when each factor of $L$ is contained in a fixed concrete bicyclic extension of $k$.