Geometric construction of metaplectic covers of \(GL_n\) in characteristic zero

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