\(\Omega\)-algebras over Henselian discrete valued fields with real closed residue field. (Q2455040)

Let \(K\) be a field of characteristic not \(2\), and let \(K(x)\) be the rational function field in one variable over \(K\), considered as the function field of the projective line \(\mathbb{P}_K^1\), so that the closed points of \(y\in\mathbb{P}_K^1\) correspond to discrete \(K\)-valuations of \(K(x)\) with residue field \(K(y)\). There is a well-known exact sequence of cohomology groups due to Fadeev which, restricted to the \(2\)-components of the respective groups, reads \[ 0\to{_2\text{Br}(K)}\to{_2\text{Br}(K(x))}@>\oplus\partial_y>>\bigoplus_{y\in\mathbb{P}_K^1} H^1(K(y),\mathbb{Z}/2\mathbb{Z})@>\sum\text{cor}>>H^1(K,\mathbb{Z}/2\mathbb{Z})\to 0, \] where \(\partial_y\) denotes the ramification map at the point \(y\), and \(\sum\text{cor}\) is the sum of the corestriction maps \(H^1(K(y),\mathbb{Z}/2\mathbb{Z})\to H^1(K,\mathbb{Z}/2\mathbb{Z})\). By Merkurjev's theorem, any \(A\in{_2\text{Br}(K(x))}\) is Brauer equivalent to a tensor product of quaternion algebras, and an interesting question is when such an \(A\) is the Brauer class of a quaternion algebra. In the present paper, the authors focus on this question in the case where \(K=\mathbb{R}(\!(t)\!)\), the field of Laurent series in one variable over the reals. Now every algebra of exponent \(2\) over \(\mathbb{R}(\!(t)\!)(x)\) is Brauer equivalent to a product of at most two quaternion algebras, as follows readily from the fact that the quadratic extension \(\mathbb{C}(\!(t)\!)(x)\) is a \(C_2\)-field. The authors try to answer the above question in terms of what they call the ramification data of \(A\in{_2\text{Br}(K(x))}\), i.e. the ramification locus \(\text{Ram}(A)=\{y\in\mathbb{P}_K^1\mid\partial_y(A)\neq 0\}\) and its value set \(\{\partial_y(A)\mid y\in\text{Ram}(A)\}\). So the question becomes which ramification data correspond to the ramification data of a quaternion algebra, and can one describe a quaternion algebra \(A\) using its ramification data by explicitly constructing a quadratic splitting field for \(A\) in terms of the ramification data. The authors provide a series of results that give partial answers to these questions. Particularly complete results are obtained for algebras that split over the real closures with respect to all orderings. The authors call such algebras \(\Omega\)-algebras. Such \(\Omega\)-algebras of exponent \(2\) over \(\mathbb{R}(\!(t)\!)(x)\) are always of index \(2\) as follows readily from a result by \textit{K. J. Becher}, who proved that the \(u\)-invariant of that field is \(4\) [Arch. Math. 86, No. 1, 31-35 (2006; Zbl 1093.11023)]. As an application, the authors use their results to derive information on conic bundle surfaces over \(\mathbb{R}(\!(t)\!)\).