The Tate-Shafarevich groups of multinorm-one tori

TL;DR

This paper investigates the Tate-Shafarevich groups associated with multinorm-one tori over global fields, providing explicit computations that relate to obstructions in local-global principles and weak approximation.

Contribution

It explicitly computes the Tate-Shafarevich groups of the character groups of multinorm-one tori for products of cyclic extensions, advancing understanding of their arithmetic properties.

Findings

Computed the Tate-Shafarevich groups for these tori.

Identified obstructions to local-global principles.

Analyzed implications for rational points and weak approximation.

Abstract

Let k be a global field and L be a finite dimensional \'etale algebra over k. In this paper, we assume that L is a product of cyclic extensions of k. Let T_{L/k} be the multinorm-one torus defined by the multinorm equation: N_{L/k} (t) = 1. Let X_c be the variety defined by the equation N_{L/k} (t) = c, for some c in k*. In this paper, we compute the Tate-Shafarevich group and the algebraic Tate-Shafarevich group of the character group of T_{L/k}. These groups measure the obstruction to the local-global principle for existence of rational points of X_c and the obstruction to the weak appraximation.