Local-global questions for tori over \(p\)-adic function fields (Q2804215)

Let \(K\) be the function field of a curve over a \(p\)-adic field. The paper under review mainly addresses local-global principles for torsors under tori defined over \(K\). Although the cohomological dimension of \(K\) is strictly greater than 2, it has been shown in recent works, especially [\textit{J.-L. Colliot-Thélène} et al., Comment. Math. Helv. 87, No. 4, 1011--1033 (2012; Zbl 1332.11065)] and [\textit{J. L. Colliot-Thélène} et al., Trans. Am. Math. Soc. 368, No. 6, 4219--4255 (2016; Zbl 1360.11068)], that when all the discrete valuations of \(K\) are considered, local-global results closely analogous to the classical ones (over number fields) may continue to hold. In the present work, the authors restrict themselves to discrete valuations that are trivial on the constant field of \(K\) and study the associated Tate-Shafarevich groups \font\fontWCA=wncyr10 {\fontWCA SH}\({}^i(T)\), \(i=1,\,2\), for algebraic tori \(T\) over \(K\).NEWLINENEWLINEThe local completions \(K_v\), which some people call 2-local fields, have very nice cohomological properties. From local duality results the authors utilize all necessary cohomological machineries to obtain the following duality theorem:NEWLINENEWLINEThere is a perfect pairing of finite groups NEWLINE\[NEWLINE \text{\font\fontWCA=wncyr10 {\fontWCA SH}}^1(T)\times \text{\font\fontWCA=wncyr10 {\fontWCA SH}}^2(T')\longrightarrow \mathbb{Q}/\mathbb{Z} NEWLINE\]NEWLINE where \(T'\) denotes the dual torus of \(T\).NEWLINENEWLINEThis duality theorem then leads to proofs that Sansuc's (by now classical) results over number fields have analogues over \(K\). Given a torsor \(Y\) under the torus \(T\), a description of the obstruction to the local-global principle for rational points on \(Y\) is given in terms of a map defined on a subquotient of \(H^3(Y,\,\mathbb{Q}/\mathbb{Z}(2))\), and this obstruction is shown to be the only one.NEWLINENEWLINEIn the last section of the paper is proved a generalization to torsors under certain reductive groups \(G\). There the simply connected cover \(G^{sc}\) of the derived group of \(G\) is assumed to be quasi-split without \(E_8\) factor. The proof relies on facts about the Rost invariant of \(G^{sc}\).NEWLINENEWLINEThe aforementioned duality theorem also has a variant for finite commutative groups (in places of tori), as stated in Theorem\;4.4.NEWLINENEWLINEThe interested readers may be referred to more recent works of \textit{D. Izquierdo} [Math. Z. 284, No. 1--2, 615--642 (2016; Zbl 1407.11130)] for variants and generalizations over similar fields, which many have even higher cohomological dimensions.