On Kernel of the regulator map

TL;DR

This paper constructs an infinitesimal form of the regulator map using methods from Bloch, Green, and Griffiths, and identifies its kernel as a specific differential form, providing insight into a related open question.

Contribution

It introduces an infinitesimal version of the regulator map and determines its kernel, advancing understanding of the regulator map's properties at the infinitesimal level.

Findings

Kernel of the infinitesimal regulator map is $ abla^{1}_{ ext{inf}}$

Supports the reasonableness of Question 1.1 at the infinitesimal level

Provides a new perspective on the structure of the regulator map

Abstract

By using the infinitesimal methods due to Bloch, Green and Griffiths, we construct an infinitesimal form of the regulator map and verify that its kernel is $\Omega^{1}_{\mathbb{C}/\mathbb{Q}}$, which suggests Question 1.1 seems reasonable at the infinitesimal level.