Adeles模型论与数论

摘要：

本文综述了阿德勒的模型论及其在模型论、代数和数论中的应用。第1-12节关注的模型论的adeles和结果是联合工程与安格斯麦金太尔。所涵盖的主题包括量词消除丰富布尔代数，量词消除限制产品和在adeles和adele空间的代数簇在自然语言中，可定义的子集adeles及其措施，解决问题的Ax从1968年的决定性环$\mathbb{Z}/m\mathbb{Z}$的所有$m>1$，可定义极小幂等元集（或“数域的素数”），稳定嵌入和第二类树性质的稳定性理论概念，adele环的初等等价和同构，公理环基本等价于限制产品和adeles，逆费费曼-沃特定理，语言adeles相关的希尔伯特符号在数论中，在adeles，和空间adele类。第13-18节是关于zeta积分和$L$函数与数论的联系。受Adele的模型论的启发，本文在Tate的论文$GL_1$和Jacquet-Langlands的论文$GL_2$的基础上，提出了一种自守形式的模型论方法，并提出了一些概念、问题和疑问。主要思想是制定概念的可构造的adelic积分和观察到的积分泰特和Jacquet-Langlands是可构造的。这些可构造的积分与模型论中的$p$ -adic和motivic积分有关.

英文摘要：

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.

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