Geometric construction of metaplectic covers of $\GL_{n}$ in characteristic zero

TL;DR

This paper introduces a new geometric method for constructing metaplectic covers of _{n} over number fields, providing explicit cocycles, splitting conditions, and applications to the power reciprocity law without using class field theory.

Contribution

It offers a novel geometric construction of metaplectic covers that avoids class field theory and algebraic K-theory, using techniques from the geometry of numbers.

Findings

Explicit 2-cocycle for _{n} over number fields

Splitting of cocycle on _{n}(k) explicitly determined

Power reciprocity law derived as a corollary

Abstract

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 $\GL_{n}(\A)$ 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.