Beyond Grothendieck's conjecture on Galois groups and arithmetic fundamental groups

超越格洛腾迪克关于伽罗瓦群和算术基本群的猜想

• 批准号: EP/T031816/1
• 负责人: Mohamed Saidi
• 金额: $ 47.15万
• 依托单位: UNIVERSITY OF EXETER
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT031816%2F1
• 关键词: Beyond; Grothendieck; conjecture; Galois; groups

The pioneering work of Galois, on the solvability by radicals of polynomials equations, established group theory at the heart of modern mathematics. The philosophy inspired by the work of Galois is that given a set of algebraic polynomial equations, various natural symmetries of the set of solutions of these equations encode some important intrinsic properties of the equations. This philosophy was empowered by the far reaching vision of Grothendieck in the second half of the last century. Grothendieck conjectured that finitely generated fields can be reconstructed quite naturally from certain groups arising as symmetry groups of solutions of certain algebraic polynomial equations; the so-called absolute Galois groups. More generally a certain class of algebraic varieties; these are mathematical objects defined by algebraic polynomial equations, can be reconstructed quite naturally from arithmetic fundamental groups arising as symmetry groups of solutions of certain algebraic polynomial equations. The vision of Grothendieck concretised in the theorems of Neukirch, Uchida, Pop, Tamagawa, and Mochizuki. They established the main foundational results of the so-called anabelian geometry and its birational version. Unfortunately, Galois groups of finitely generated fields, and likewise arithmetic fundamental groups, are still very mysterious objects. A full and explicit understanding of these objects seems to be out of reach in the foreseeable future. Class field theory provides an explicit description of some rather small portion of Galois groups and arithmetic fundamental groups: the abelian quotients. The main objective of this proposal is to establish new results in this area of mathematical research, whereby one can naturally reconstruct finitely generated fields, as well as certain algebraic varieties, from some quotients of Galois groups and arithmetic fundamental groups; the so-called m-step solvable quotients, which are better understood. These quotients are built up successively, step by step, starting from abelian quotients which are rather well understood by class field theory. Such results would be a substantial sharpening of the foundational results of the theory, and would pave the way to a more explicit, and applicable, Galois theory of finitely generated fields and algebraic varieties.

伽罗瓦的开创性工作，关于多项式方程根的可解性，建立了现代数学的核心群论。伽罗瓦的工作所启发的哲学是，给定一组代数多项式方程，这些方程的解的集合的各种自然对称性编码了方程的一些重要的内在性质。这种哲学在上个世纪后半叶被格罗滕迪克的深远愿景所授权。Grothendieck指出，生成的场可以很自然地从某些代数多项式方程的解的对称群中产生的某些群中重建出来，即所谓的绝对伽罗瓦群。更一般地说，某一类代数簇;这些是由代数多项式方程定义的数学对象，可以很自然地从作为某些代数多项式方程的解的对称群而产生的算术基本群中重建出来。 格罗滕迪克的愿景具体体现在纽基奇、内田、波普、玉川和Mochizuki的定理中。他们建立了所谓的阿纳贝尔几何及其双有理版本的主要基础结果。不幸的是，伽罗瓦群和算术基本群仍然是非常神秘的对象。在可预见的将来，对这些物体的全面和明确的理解似乎是遥不可及的。类域理论提供了伽罗瓦群和算术基本群的一小部分的明确描述：阿贝尔群。这个提议的主要目的是在这一数学研究领域建立新的成果，人们可以自然地从伽罗瓦群和算术基本群的某些子元中重建n阶生成域以及某些代数簇;所谓的m阶可解子元，这是更好地理解的。这些等价式是从类场理论相当好地理解的阿贝尔等价式开始，一步一步地连续建立起来的。这样的结果将是一个实质性的锐化的基础结果的理论，并铺平了道路，以一个更明确的，和适用的伽罗瓦理论的代数生成领域和代数簇。

项目成果

m-step solvable Hom-form of the birational Grothendieck conjecture for number fields

数域双有理格洛腾迪克猜想的 m 步可解 Hom 形式

• 发表时间: 2023
• 作者: Alberto Corato

The m-step solvable anabelian geometry of finitely generated fields in positive characteristics

正特性有限生成场的 m 步可解阿贝尔几何

• 作者: Mohamed Saidi

The m-step solvable anabelian geometry of finitely generated fields in characteristic zero

特征零条件下有限生成场的 m 步可解阿贝尔几何

• 作者: Mohamed Saidi

The m-step solvable anabelian geometry of number fields

数域的 m 步可解阿贝尔几何

• 期刊: Journal fur die Reine und Angewandte Mathematik
• 影响因子: 1.5
• 作者: Mohamed Saidi

The m-step solvable anabelian geometry of algebraic function fields

代数函数域的 m 步可解阿贝尔几何

• 作者: Mohamed Saidi