%0 Journal Article
%T Geometric construction of metaplectic covers of $GL_n$ in characteristic zero
%A Richard Hill
%J Online Journal of Analytic Combinatorics
%D 2010
%I University of Auckland
%X This paper presents a new construction of the $m$-fold metaplectic cover of $GL_n$ over an algebraic number field $k$, where $k$ contains a primitive $m$-th root of unity. A 2-cocycle on $GL_n(mathbb A)$ representing this extension is given and the splitting of the cocycle on $GL_n(k)$ is found explicitly. The cocycle is smooth at almost all places of $k$. As a consequence, a formula for the Kubota symbol on $SL_n$ is obtained. The construction of the paper requires neither class field theory nor algebraic K-theory, but relies instead on naive techniques from the geometry of numbers introduced by W. Habicht and T. Kubota. The power reciprocity law for a number field is obtained as a corollary.
%U http://analytic-combinatorics.org/index.php/ojac/article/view/67