Local-global principles for norm one tori over semi-global fields (Q2031695)

Semi-global fields are function fields of curves over complete discretely valued fields. Examples include \(\mathbb{Q}_{p}(x)\) and \(\mathbb{C}((t))(x)\). The field patching technique of Harbater-Hartmann-Krashen has been used to show that over a semi-global field \(F\), local-global principles for torsors under \(F\)-rational linear algebraic groups hold if the group is connected or the reduction graph associated to a two-dimensional normal projective model of \(F\) is a tree [\textit{D. Harbater} et al., Am. J. Math. 137, No. 6, 1559--1612 (2015; Zbl 1348.11036)]. It turns out that the rationality hypothesis on the group cannot be dispensed with in general. The authors in [\textit{J. Colliot-Thélène} et al., Trans. Am. Math. Soc. 368, No. 6, 4219--4255 (2016; Zbl 1360.11068)] have shown that local-global principles do not hold for torsors under a norm one torus associated to a product of three quadratic extensions. The underlying residue field of the semi-global field in their example is assumed to be separably closed. In a positive direction, the paper under review proves a local-global principle for norm one tori associated to Galois field extensions of a semi-global field \(F\). Let \(L/F\) be a Galois extension with degree \(n\) coprime to the characteristic of the underlying residue field \(k\). Assume that \(k\) is algebraically closed or a finite field containing a primitve \(n^{th}\)-root of unity. Suppose further that the reduction graph of a two-dimensional regular proper model of \(F\) is a tree. The author shows that local-global principle with respect to all discrete valuations of \(F\) holds for torsors under the norm one torus associated to \(L/F\). An important step consists in showing that local-global principle holds with respect to a finite collection of overfields considered in the field patching set-up (see [\textit{D. Harbater} et al., Invent. Math. 178, No. 2, 231--263 (2009; Zbl 1259.12003)]). To establish this in our situation, one needs to show that at the branch fields, the field valued points of the torus can be ``simultaneously factorized'' [\textit{D. Harbater} et al., Am. J. Math. 137, No. 6, 1559--1612 (2015; Zbl 1348.11036)]. The author uses a description of \(R\)-trivial elements of such tori due to \textit{J.-L. Colliot-Thelene} and \textit{J.-J. Sansuc} [Ann. Sci. Éc. Norm. Supér. (4) 10, 175--229 (1977; Zbl 0356.14007), Proposition 15]. The assumption on the residue field and the above-mentioned description of \(R\)-trivial elements enables him to show that at the branch fields, every element in the torus can be written as product of an \(R\)-trivial element and an \(n^{th}\)-root of unity. This is then used to prove that the simultaneous factorization property for the torus at the branch fields holds. The hypothesis on the reduction graph is needed to deal with simultaneous factorization for the term lying in the group of roots of unity. Following this, the author shows that local-global principle also holds with respect to discrete valuations of \(F\). He also provides counterexamples indicating that local-global principle for torsors under norm one tori of Galois extensions do not hold when the reduction graph is not a tree.