Computing Borel's Regulator

TL;DR

This paper introduces a new infinite series formula for Borel's regulator using the Karoubi-Hamida integral, and explicitly constructs elements in K_3 of cyclotomic fields to compute their regulator values.

Contribution

It provides a novel approach to compute Borel's regulator for cyclotomic fields by explicitly constructing elements in K_3 and evaluating their images using a new formula.

Findings

Derived an infinite series formula for the Borel class.

Explicitly constructed elements in K_3(F) for cyclotomic fields.

Computed Borel regulator values for specific fields.

Abstract

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL(\mathbb{C})). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H_{3}(GL(\mathbb{C})) under the Hurewicz map. Applying our formula to these images yields a value V_1(F), which coincides with the Borel regulator R_1(F) when our set is a basis of K_3(F) modulo torsion. For F= \mathbb{Q}(e^{2\pi i/3}) a computation of V_1(F) has been made based on our techniques.