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Model Theory of Adeles and Number Theory

This paper is a survey on model theory of adeles and applications to model theory, algebra, and number theory. Sections 1-12 concern model theory of adeles and the results are joint works with Angus Macintyre. The topics covered include quantifier elimination in enriched Boolean algebras, quantifier elimination in restricted products and in adeles and adele spaces of algebraic varieties in natural languages, definable subsets of adeles and their measures, solution to a problem of Ax from 1968 on decidability of the rings $\mathbb{Z}/m\mathbb{Z}$ for all $m>1$, definable sets of minimal idempotents (or "primes of the number field" ) in the adeles, stability-theoretic notions of stable embedding and tree property of the second kind, elementary equivalence and isomorphism for adele rings, axioms for rings elementarily equivalent to restricted products and for the adeles, converse to Feferman-Vaught theorems, a language for adeles relevant for Hilbert symbols in number theory, imaginaries in adeles, and the space adele classes. Sections 13-18 are concerned with connections to number theory around zeta integrals and $L$-functions. Inspired by our model theory of adeles, I propose a model-theoretic approach to automorphic forms on $GL_1$ (Tate's thesis) and $GL_2$ (work of Jacquet-Langlands), and formulate several notions, problems and questions. The main idea is to formulate notions of constructible adelic integrals and observe that the integrals of Tate and Jacquet-Langlands are constructible. These constructible integrals are related to the $p$-adic and motivic integrals in model theory.