Langlands reciprocity for C*-algebras

TL;DR

This paper constructs a $C^*$-algebraic framework for Langlands reciprocity, linking the $K$-theory of algebra inclusions to automorphic $L$-functions of varieties over number fields.

Contribution

It introduces $C^*$-algebras for varieties and groups, and establishes an embedding that generalizes Langlands reciprocity in the $C^*$-algebra setting.

Findings

The $K$-theory of the algebra inclusion relates the Hasse-Weil $L$-function to automorphic $L$-functions.

An embedding of the algebra of a variety into that of a reductive group is constructed.

The framework applies to $G$-coherent varieties like Shimura varieties.

Abstract

We introduce a $C^*$-algebra $\mathscr{A}_V$ of a variety $V$ over the number field $K$ and a $C^*$-algebra $\mathscr{A}_G$ of a reductive group $G$ over the ring of adeles of $K$. Using Pimsner's Theorem we construct an embedding $\mathscr{A}_V\hookrightarrow \mathscr{A}_G$, where $V$ is a $G$-coherent variety, e.g. the Shimura variety of $G$. The embedding is an analog of the Langlands reciprocity for $C^*$-algebras. It follows from the $K$-theory of the inclusion $\mathscr{A}_V\subset\mathscr{A}_G$ that the Hasse-Weil $L$-function of $V$ is a product of the automorphic $L$-functions corresponding to irreducible representations of the group $G$.