A Note on Number Fields Sharing the List of Dedekind Zeta-Functions of Abelian Extensions with some Applications towards the Neukirch-Uchida Theorem

Imaginary quadratic fields with isomorphic abelian Galois groups

In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a classification by Kubota of idele class character groups in terms of their infinite families of Ulm invariants, and did not yield a description of $A_K$. In this paper, we provide a direct `computation' of the profinite group $A_K$ for imaginary quadratic $K$, and use it to obtain many differe…

由某些交叉积C*-代数的*-同构构造数域同构

Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras

本研究证明了与数域乘法群作用于有限阿代尔环相关的交叉积C*-代数的刚性，即由这类代数的*-同构可构造对应数域的同构，并借此用拓扑全群得到Neukirch–Uchida定理的类似结果，给出一类可完全刻画数域的离散群。

The $m$-step solvable anabelian geometry of number fields

Given a number field $K$ and an integer $m\geq 0$, let $K_m$ denote the maximal $m$-step solvable Galois extension of $K$ and write $G_K^m$ for the maximal $m$-step solvable Galois group Gal$(K_m/K)$ of $K$. In this paper, we prove that the isomorphy type of $K$ is determined by the isomorphy type of $G_K^3$. Further, we prove that $K_m/K$ is determined functorially by $G_K^{m+3}$ (resp. $G_K^{m+4}$) for $m\geq 2$ (resp. $m \leq 1$). This is a substantial sharpening of a famous theorem of Neukirch and Uchida. A key step in our proof is the establishment of the so-called local theory, which in …

The arithmetic local Nori fundamental group

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field $k$. We give particular attention to the case of fields: to any field extension $K/k$ we attach a pro-local group scheme over $k$. We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of t…

An idelic view of ideals

Ideles and adeles can be viewed as a generalization of Minkowski theory, in which embedding of a number field to the Cartesian product of its completions at the archimedean valuation is generalized to an embedding of the Cartesian product of all its completions with some restriction. This paper introduces the basic notions of point-set topology and builds the real numbers from the rational numbers. Then we review concepts from local fields that will lead to the product formula and the approximation theorem. We, then, construct adeles and ideles. The ideles modulo $k^*$ maps surjectively to the…

Quantum Statistical Mechanics, L-series and Anabelian Geometry

It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C*-algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendi…