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

Let K be a complete discretely valued field with the residue field κ. Assume that the cohomological dimension of κ is less than or equal to 1 (for example, κ 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(κ). 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 \({\mathscr X}\) $$\mathcal{X}$$ 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 κ is algebraically closed with char(κ) ≠ 2. If the graph associated to \({\mathscr X}\) $$\mathcal{X}$$ 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 the others are of degree not divisible by 4.