Defining $\mathbb{A}$ in $G(\mathbb{A})$

TL;DR

This paper explores the logical interpretability of adele rings in relation to Chevalley groups, establishing a bi-interpretability connection that enhances understanding of their structural relationship.

Contribution

It demonstrates that for Chevalley groups, the adele ring is bi-interpretable with the group over that ring, revealing a deep logical equivalence.

Findings

Adele rings are bi-interpretable with Chevalley groups over them.

Establishes a logical equivalence between algebraic and ring-theoretic structures.

Provides a framework for understanding the definability of algebraic groups in number theory.

Abstract

If $G$ is a Chevalley group and $R$ is an adele ring, or a product of local factors in an adele ring, then $R$ is bi-interpretable with $G(R)$.