Model Theory of Adeles I

TL;DR

This paper explores the model-theoretic properties of the ring of adeles of a number field, achieving quantifier elimination and analyzing definable sets and measures within this structure.

Contribution

It provides the first quantifier elimination results for the adele ring in the language of rings with enrichments, advancing understanding of its logical structure.

Findings

Quantifier elimination in the language of rings for adeles

Characterization of definable subsets of adeles

Analysis of measures of definable sets

Abstract

We study the model theory of the ring of adeles of a number field. We obtain quantifier elimination results in the language of rings and some enrichments. We given consequences for definable subsets of the adeles, and their measures.