Local-global principles for multinorm tori over semi-global fields

Let $K$ be a complete discretely valued field with the residue field $\kappa$. Assume that cohomological dimension of $\kappa$ is less than or equal to $1$ (for example, $\kappa$ is an algebraically closed field or a finite field). Let $F$ be the function field of a curve over $K$. Let $n$ be a squarefree positive integer not divisible by char$(\kappa )$. Then for any two degree $n$ abelian extensions, we prove that the local-global principle holds for the associated multinorm torus with respect to discrete valuations. Let $\mathcal{X}$ be a regular proper model of $F$ such that the reduced special fibre $X$ is a union of regular curves with normal crossings. Suppose that $\kappa$ is algebraically closed with $char(\kappa )\mathrm{\ne }2$. If the graph associated to $\mathcal{X}$ is a tree (e.g. $F=K(t)$) then we show that the same local-global principle holds for the multinorm torus associated to finitely many abelian extensions where one of the extensions is quadratic and others are of degree not divisible by $4$.