Geometric construction of metaplectic covers of $G{L}_n$ in characteristic zero

Abstract

This paper presents a new construction of the $m$-fold metaplectic cover of ${\mathrm{G}\mathrm{L}}_n$ over an algebraic number field $k$, where $k$ contains a primitive $m$-th root of unity. A 2-cocycle on ${\mathrm{G}\mathrm{L}}_n(\mathbb{A})$ representing this extension is given, and the splitting of the cocycle on ${\mathrm{G}\mathrm{L}}_n(k)$ is found explicitly. The cocycle is smooth at almost all places of $k$. As a consequence, a formula for the Kubota symbol on ${\mathrm{S}\mathrm{L}}_n$ 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.