On the automorphic side of the K-theoretic Artin symbol

TL;DR

This paper explores the automorphic aspect of the K-theoretic Artin symbol, providing proofs for Clausen's predictions by connecting Galois and automorphic sides through K-theory and homotopical methods.

Contribution

It offers new proofs and insights into the automorphic side of the K-theoretic Artin symbol, bridging homotopical enrichment with classical automorphic objects.

Findings

Proofs for Clausen's predictions on the automorphic side

Connection established between Selmer K-homology and automorphic K-theory

Enhanced understanding of the homotopical enrichment in class field theory

Abstract

Clausen has constructed a homotopical enrichment of the Artin reciprocity symbol in class field theory. On the Galois side, Selmer K-homology replaces the abelianized Galois group, while on the automorphic side the K-theory of locally compact vector spaces replaces classical idelic objects. We supply proofs for some predictions of Clausen regarding the automorphic side.