Reciprocity for Kato-Saito idele class group with modulus

TL;DR

This paper constructs a reciprocity homomorphism linking the Kato-Saito idele class group with modulus to an étale fundamental group with modulus, advancing the understanding of class field theory in algebraic geometry.

Contribution

It introduces a new reciprocity homomorphism connecting Kato-Saito idele class groups with a fundamental group, providing a K-theoretic perspective and unifying existing concepts.

Findings

Established a reciprocity homomorphism for the Kato-Saito idele class group with modulus.

Connected the K-theoretic and cycle-theoretic idele class groups.

Provided a new interpretation of the étale fundamental group with modulus.

Abstract

We introduce an etale fundamental group with modulus and construct a reciprocity homomorphism from the Kato-Saito idele class group with modulus to this fundamental group. This is the K-theoretic analogue of the reciprocity for the cycle-theoretic idele class group with modulus due to Kerz-Saito, and plays a central role in showing the isomorphism between the two idele class groups. It also provides a new interpretation of the already known etale fundamental group with modulus due to Deligne and Laumon.