Torsors over the punctured affine line (Q2851606)

Let \(k\) be a field with fixed separable closure \(k_s\), and \(G\) be a reductive group over \(k\). According to a well-known theorem of \textit{M. S. Ragunathan} and \textit{A. Ramanathan} [Proc. Indian Acad. Sci., Math. Sci. 93, 137--145 (1984; Zbl 0587.14007)], the natural map of Galois cohomology sets \(H^1(k, G)\to H^1(k, G(\mathbf{A}^1_{k_s}))\) is bijective, where \(\mathbf{A}^1_{k}\) is the affine line over \(k\). In other words, every \(\mathbf{A}^1_{k}\)-torsor under \(G\) split by a separable base extension is constant. In the present paper the authors study the analogous question for \(\mathbf{A}^\times_{k}:=\mathbf{A}^1_{k}\setminus\{0\}\). It is not true that \(\mathbf{A}^\times_{k}\)-torsors split by a separable base extension are constant, even for \(G\) semisimple. Instead, the authors prove the following theorem: for \(G\) reductive, the natural map of cohomology sets \(H^1(\mathbf{A}^\times_{k}, G)\to H^1(k((t)), G)\) is bijective, provided that the characteristic of \(k\) is not among a finite set of bad primes associated with \(G\) using the structure theory of reductive groups. Under this assumption \(\mathbf{A}^\times_{k}\)-torsors under \(G\) are split by a separable base extension, so that the former set can be identified with \(H^1(k, G(\mathbf{A}^\times_{k_s}))\), and the latter with \(H^1(k, G(k((t))_{nr})\), where the subscript \(nr\) denotes the maximal unramified extension. Since the set \(H^1(k((t)), G)\) can be determined using Bruhat-Tits theory, this yields a classification of \(\mathbf{A}^\times_{k}\)-torsors under \(G\).NEWLINENEWLINEThe authors' approach differs at several points from that of Ragunathan and Ramanathan, and they include a proof of the original result over \(\mathbf{A}^1_{k}\) using their method. The key technical point is to establish the existence of certain maximal tori in reductive groups, which is done here based on building-theoretic results due to Soulé in the case of the affine line and to Abramenko in the case of \(\mathbf{A}^\times_{k}\).