Metaplectic extensions of quaternion skew fields
The author considers central extensions of \(D_{\mathbb A}^{\times}\) where \(D\) is a quaternion algebra over a number field \(k\). The extension is assumed to be a ``product'' over the places of \(k\) and to split over \(D^{\times}\). In a previous paper [ibid. 191, 261--282 (1986; Zbl 0563.12014)] the author has studied the corresponding local problem. There are two obvious constructions of cocycles with values in \(\mu_ n(k)\)- one by restriction of the Kubota cocycle from \(\mathrm{GL}_ 2(k_{\mathbb A}')\) where \(k'\) is a splitting field and the other by lifting back a cocycle of \(\mathrm{GL}_ 1(k_{\mathbb A})\) by the reduced norm. The author shows that all central topological extensions by \(\mu_ n(k)\) are generated by these when \(n\) is odd. The proof is based on his previous characterization of the local extensions by commutators and a version of Moore's theorem. Also included are some remarks on the case n being a power of 2, which is more complicated.
scientific article; zbMATH DE number 3981289