This paper addresses certain arithmetic problems over the function field \(K\) of a smooth projective curve \(X\) over a higher local field \(k\). Here higher local fields are defined inductively: A \(d\)-local field is a complete discrete valuation field whose residue field is \((d-1)\)-local, the \(0\)-local fields being finite fields or fields of the form \(\mathbb{C}((t))\). In previous work ([J. Number Theory 157, 250–270 (2015; Zbl 1367.11045)] and [Math. Z. 284, No. 1–2, 615–642 (2016; Zbl 1407.11130)]), the author obtained arithmetic duality theorems for the function field \(K\).
In this paper, he applies his previous results to study weak approximation and local-global principle questions for algebraic tori \(T\) over \(K\). It seems not easy to fully state the main results in a brief way. But there are two interesting exact sequences which describe the closures of the rational points \(T(K)\) in the products \(\prod_{v\in X^{(1)}}T(K_v)\) and \(\prod_{v\in S}T(K_v)\), where \(X^{(1)}\) is the set of all closed points of the curve \(X\) and \(S\) is a finite subset of \(X^{(1)}\). The author also characterizes the weak approximation property of \(T\) by using Tate-Shafarevich groups of certain motivic complexes associated to the torus \(T\). Two weaker versions of weak approximation are studies as well. The main results the paper include necessary and sufficient conditions for these versions of weak approximation.