Fredholm modules and the Beilinson-Bloch regulator

TL;DR

This paper provides an operator-theoretic reconstruction of the Beilinson-Bloch regulator for compact Riemann surfaces, utilizing Fredholm modules, loop operators, and the Connes-Karoubi character, with new computational insights.

Contribution

It introduces a novel proof of the Beilinson-Bloch regulator using operator theory and computes the Connes-Karoubi character for Steinberg symbols on the circle without relying on joint torsion theory.

Findings

Reconstruction of the Beilinson-Bloch regulator via operator theory

New computation of the Connes-Karoubi character for Steinberg symbols

Application of Helton-Howe determinant theory in this context

Abstract

We prove an operator-theoretic reconstruction of the Beilinson-Bloch regulator for compact Riemann surfaces, using loop operators and the Connes-Karoubi character for Fredholm modules. The proof includes a new computation of the Connes-Karoubi character for Steinberg symbols of the circle, which relies on the Helton-Howe determinant theory, but not on the Carey-Pincus theory of joint torsion.