Model theory of absolute Galois groups with a view towards arithmetic geometry

格罗森迪克在他写给法尔廷的著名信件中勾勒出后来被称为阿纳贝尔几何学的程序。本着这个程序的精神，我们致力于分析绝对Galois群的模型理论，特别是数域的绝对Galois群的模型理论。域K的绝对Galois群GK是K的代数闭包的自同构群，它以元素形式确定K。Gk是K的一个重要不变量；人们可以认为这个群在场的上下文中类似于基本群。一旦我们引入所谓的故事拓扑学，这实际上不仅仅是一个类比。通常，由于伽罗华上同调的机制，GK以计算可访问的方式对关于K的算术信息进行编码。更具体地说，我们将研究关于有理数的域Q的绝对伽罗瓦群的两个猜想：绝对伽罗瓦猜想(AGC)：任何域K的绝对伽罗华群同构于Q的绝对伽罗华群同构于Q的绝对伽罗瓦群具有亨氏赋值与可除值群，且具有与Q同构的剩余域；和初等伽罗瓦猜想(EGC)，即任何域K的绝对伽罗华群与Q的绝对伽罗华群初等等价，它具有亨赛尔赋值与可除值群，且剩余域与Q基本等价。不难看出，EGC蕴含着AGC，并且Koenigsmann的一个非平凡定理是AGC等价于Grothendieck回转几何中的所谓二元截面猜想。接近这些猜想的技术包括一般赋值理论，即研究绝对Galois群的结构，重点是它们如何看到估值(这已经由Koenigsmann的早期工作很好地发展了)，菲利普·迪特曼(Philip Dittmann)在绝对伽罗华群模型理论方面的最新进展，从代数数论(包括伽罗华上同调)中精炼出的技术，以及可能来自稳定性理论的一些输入。该项目结合了EPSRC的三个研究领域：代数、逻辑/组合学和数论。

In his famous letter to Faltings, Grothendieck sketched the programme which came to be known as anabelian geometry. It is in the spirit of this programme that we endeavour to analyse the model theory of absolute Galois groups, in particular of absolute Galois groups of number fields.The absolute Galois group GK of a field K is the group of automorphisms of the algebraic closure of K that fix K elementwise. GK is an important invariant of K ; one may think this group as being the analogue of a fundamental group in the context of fields. This is actually more than a mere analogy once we introduce the so called étale topology. Typically GK encodes arithmetic information about K in a computationally accessible manner, due to the machinery of Galois cohomology. This gives rise to the hope for effective versions of Faltings' celebrated Theorem about the finiteness of the number of rational points on algebraic curves of genus >1 over number fields.More concretely, we will work on two conjectures about the absolute Galois group of the field Q of rational numbers: the Absolute Galois Conjecture (AGC) that any field K whose absolute Galois group is isomorphic to that of Q carries a henselian valuation with divisible value group and with residue field isomorphic to Q; and the Elementary Galois Conjecture (EGC) that any field K whose absolute Galois group is elementarily equivalent to that of Q carries a henselian valuation with divisible value group and with residue field elementarily equivalent to Q. It is not hard to see that EGC implies AGC and it is a nontrivial theorem of Koenigsmann that AGC is equivalent to the so-called birational section conjecture in Grothendieck's anabelian geometry.The techniques for approaching these conjectures include general valuation theory, a study of the structure of absolute Galois groups with a focus on how they "see" valuations (this is already well developed by the earlier work of Koenigsmann), recent progress on the model theory of absolute Galois groups by Philip Dittmann (2018), refined techniques from algebraic number theory including Galois cohomology and possibly some inputs from stability theory.The project combines the three EPSRC research areas: Algebra, Logic/Combinatorics and Number Theory.