Geometric structures over the absolute point and their arithmetic

具有相关算术内容的“绝对”几何理论的发展代表了基础研究的一个新方向，它将代数几何、非对易几何、数论和数学物理等领域相互联系起来。作为正特征整体场结构基础的代数曲线几何在解决数论中几个基本问题的过程中发挥了至关重要的作用，这些基本问题对特征为零的整体场仍然是开放的。一些组合公式，如提供有限基数域Q上的Grassman有理点集的基数的方程，当Q=1时也是保持有意义值的有理表达式。A.Weil和K.Iwaawa的经典观点是邻接单位根是一个类似于构造基场扩张的过程，当伽罗华域的基数(Q)趋于1时，也激发了对预期为伽罗华域的非平凡极限的数学对象的搜索。已展示了在Hasse-Weil Zeta函数中将极限Q取为1的过程，以确定作为Riemann Zeta函数在此基本框架中的几何对应物的“绝对曲线”的“计数函数”N(Q)。此外，基于Bost-Connes量子统计力学系统(最初给出了非对易几何与数论之间的关系)和相应的几何空间(即有理数域的Adele类空间)，计数函数N(Q)被解释为与Riemann-Weil显式公式和L函数零点的谱解释密切相关的交数。本研究的目的是揭示Adele类空间及其变体的几何，并将其与绝对曲线进行比较，利用代数几何、非对易几何、数论(包括岩泽理论)和热带几何。拟议的项目致力于塑造一种新的几何结构的结构和促进对其基本性质的研究，目的是将数论中的A.Weil在他对函数域的Riemann假设的证明中的思想移植到代数数域的情况。所使用的方法主要来自代数几何、非对易几何、数论(包括岩泽理论)和热带几何，计算机考试是重要的工具。该项目的广泛影响是向年轻研究人员和资深科学家的受众传达需要通过致力于发展数论和非对易几何领域之间的新联系来攻击对算术中一些主要开放问题的概念性理解的需要，尽管这是一个非常新的数学领域，但在最近几年里已经迅速成熟。这项研究涉及数论中的一些关键数学对象，例如著名的Riemann Zeta函数、它的推广(L函数)和岩泽理论中的p-进对应函数。该项目的发展也有望巩固非对易几何、数论和数学之间的相互作用，使之成为一种统一的方法论“，包括热带几何。

The development of an "absolute" geometric theory with relevant arithmetic contents represents a new direction of fundamental research, interconnecting the fields of algebraic geometry, noncommutative geometry, number-theory and mathematical physics. The geometry of algebraic curves underlying the structure of global fields of positive characteristic has shown its crucial role in the process of solving several fundamental questions in number-theory which are still open for global fields of characteristic zero. Some combinatorial formulas, like the equation supplying the cardinality of the set of rational points of a Grassmannian over a finite field of cardinality q are known to be rational expressions keeping a meaningful value also when q=1. The classical point of view of A. Weil and K. Iwasawa that adjoining roots of unity is a process analogous to the construction of an extension of a base field, also motivates the search of a mathematical object that is expected to be a non-trivial limit of Galois fields as the cardinality (q) of these fields tends to 1. The process of taking the limit q to 1 in the Hasse-Weil zeta function has been shown to determine the "counting function" N(q) of the "absolute curve" which is the geometric counterpart, in this basic framework, of the Riemann zeta function. Moreover, based on the Bost-Connes quantum statistical mechanical system (which originally gave the relation between noncommutative geometry and number theory) and on the corresponding geometric space (i.e. the adele class space of the global field of rational numbers) the counting function N(q) is interpreted as an intersection number closely related to both the Riemann-Weil explicit formulas and the spectral interpretation of the zeros of L-functions.The goal of the proposed research is to unveil the geometry of the adele class space and its variants, and to compare these spaces with the "absolute curve", using the tools of algebraic geometry, noncommutative geometry, number theory (including Iwasawa theory), and tropical geometry.The proposed project is devoted to shape the construction and promote the study of the fundamental properties of a new geometric structure with the goal to transplant the ideas of A. Weil in number-theory, in his proof of the Riemann Hypothesis for function fields, to the case of algebraic number fields. The methods to be used come mainly from algebraic geometry, noncommutative geometry, number theory (including Iwasawa theory) and tropical geometry, with computer testing as an important tool. The broad impact of the project is to communicate to an audience of young researchers and senior scientists the need to attack the conceptual understanding of some of the main open problems in arithmetic by working on the development of new connections between the fields of number theory and noncommutative geometry which although a very new area of mathematics, has matured rapidly in the recent past few years. This research addresses central aspects and open problems that involve some of the key mathematical objects in number-theory, such as the celebrated Riemann zeta function, its generalizations (L-functions) and the p-adic counterparts in Iwasawa theory. The development of this project is also expected to solidify the interactions between noncommutative geometry, number theory and mathematics in \characteristic one", including tropical geometry, under a unified methodology.