Space forms and higher metaplectic groups (Q1392409)

Although there are a large number of distinct proofs of the law of quadratic reciprocity, up to 1987 only one of these, that using the theory of genera, served as the basis for a proof of the general reciprocity law. In that year \textit{T. Kubota} [Jap. J. Math., New Ser. 13, 235-275 (1987; Zbl 0639.12004)] discovered a new proof, a generalization of the Gauss Lemma/lattice point method. This method is close in spirit to the K-theory interpretation of the reciprocity law (also due to Kubota) but is technically intricate. The point of the present paper is to try to understand the relationship between Kubota's proof and the construction of metaplectic groups (or K-theory). The author works in the context of a global field of finite characteristic; in this case one can prove the reciprocity law in an elementary fashion using resultants (Weil). What the author does is first (using an idea going back to Gauss) to construct cohomology \(\mathbb{F}^\times_q\) classes on \(G(k_v)\), \(G\) a fixed algebraic group, \(v\) a place of \(k\), defining an extension in a fairly abstract setting. The construction appears to be quite different from that of Matsumoto. Then he shows that the global extension constructed out of the local ones splits over \(G(k)\), by an argument close to Kubota's proof of the reciprocity law. It is indeed, as Kubota showed, another formulation of the reciprocity law and one which is close in spirit to K-theory and the theory of metaplectic groups.