On the Contou-Carrere Symbol for Surfaces

TL;DR

This paper introduces the Contou-Carrere symbol for surfaces, providing foundational results and explicit computations using iterated integrals over membranes, advancing understanding of this mathematical construct.

Contribution

It offers the first detailed computation of the Contou-Carrere symbol for surfaces, combining theoretical groundwork with explicit integral-based calculations.

Findings

Established technical results for defining the symbol

Computed all components of the symbol for surfaces

Utilized iterated integrals over membranes for calculations

Abstract

This is a preliminary report on the Contou-Carrere symbol for surfaces. It consists of two parts. In the first part, we recall technical results needed to define the symbol. The second part is where we compute all components of the Coutou-Carrere symbol for surfaces, using iterated integrals over membranes.