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

在他写给法尔廷斯的那封著名的信中，格罗滕迪克概述了后来被称为安娜贝尔几何的程序。正是本着这个计划的精神，我们努力分析绝对伽罗瓦群的模型理论，特别是绝对伽罗瓦群的数域。域K的绝对伽罗瓦群GK是K的代数闭包的自同构群，这些自同构群在元素上固定K。GK是K的一个重要不变量；人们可能会认为这个群是场背景下的基本群的类似物。这实际上不仅仅是一个类比，一旦我们引入了所谓的<s:1>拓扑结构。由于伽罗瓦上同调的机制，GK以一种计算上可访问的方式编码关于K的算术信息。这使人们有希望得到Faltings著名定理的有效版本，该定理是关于在数域上属bbbb1的代数曲线上有理点数目有限的。更具体地说，我们将研究关于有理数域Q的绝对伽罗瓦群的两个猜想：绝对伽罗瓦猜想（AGC），即任何域K的绝对伽罗瓦群与Q同构，其值群可除且残域与Q同构的henselian值；以及初等伽罗瓦猜想（EGC），即任何域K的绝对伽罗瓦群与Q的绝对伽罗瓦群初等等价，其值群可分且剩余域初等等价于Q的亨塞利值。不难看出，EGC隐含着AGC，而AGC等价于格罗thendieck的无伪几何中所谓的双国截面猜想是柯尼斯曼的一个非平凡定理。接近这些猜想的技术包括一般估值理论，对绝对伽罗瓦群结构的研究，重点是它们如何“看到”估值（这已经由Koenigsmann的早期工作发展得很好），Philip Dittmann在绝对伽罗瓦群模型理论方面的最新进展（2018），代数数论的改进技术，包括伽罗瓦上同调，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.