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The Tate-Shafarevich groups of multinorm-one tori

Abstract

Let k be a global field and $L$ be a product of cyclic extensions of k. Let T be the torus defined by the multinorm equation N_{L/k}(x)=1 and let \hat T be its character group. In this talk we are interested in the Tate--Shafarevich group and the algebraic Tate--Shafarevich group of \hat T. These groups give the obstructions to the Hasse principles and the weak approximations for rational points on principle homogeneous spaces of T. We give concrete description of these groups and provide several examples.