The 3-rd unramified cohomology for norm one torus

Abstract

For an algebraic torus $S$, Blinstein and Merkurjev have given an estimate of $3$-th unramified cohomology ${\bar{H}}_{nr}^3(F(S),\mathbb{Q}/\mathbb{Z}(2))$ obtained from a flasque resolution of $S$. Based on their work, for the norm one torus $W=R_{K/F}^{(1)}{\mathbb{G}}_m$ with $K/F$ abelian, we compute the $3$-th unramified cohomology ${\bar{H}}_{nr}^3(F(W),\mathbb{Q}/\mathbb{Z}(2))$.