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arXiv · 1912.09823

The Tate-Shafarevich groups of multinorm-one tori

Ting-Yu Lee
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Abstract

Let k be a global field and L be a finite dimensional étale 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.

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Ting-Yu Lee. 2021-07-13. The Tate-Shafarevich groups of multinorm-one tori. https://arxiv.org/abs/1912.09823

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