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

在他著名的信中，法尔亭，格罗滕迪克勾画了该计划后来被称为anabelian几何。正是本着本计划的精神，我们努力分析绝对伽罗瓦群的模型理论，特别是数域的绝对伽罗瓦群。域K的绝对伽罗瓦群GK是K的代数闭包的自同构群，它按元素固定K。GK是K的一个重要不变量;人们可以认为这个群是场背景下基本群的类似物。一旦我们引入所谓的etale拓扑，这实际上不仅仅是一个类比。通常GK以计算可访问的方式编码关于K的算术信息，这是由于伽罗瓦上同调机制。这就给了我们希望，找到关于数域上亏格大于1的代数曲线上有理点个数有限性的著名的Faltings定理的有效版本。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定理：绝对伽罗瓦猜想（AGC）任何域K，其绝对伽罗瓦群同构于Q的绝对伽罗瓦群，都有一个具有可除值群和剩余域的Hensel赋值初等伽罗瓦猜想（EGC）：任何域K的绝对伽罗瓦群初等等价于Q的绝对伽罗瓦群，其Hensel赋值具有可分值群，其剩余域初等等价于Q。不难看出，EGC蕴涵AGC，并且这是Koenigsmann的一个非平凡定理，AGC等价于Grothendieck的Anabel几何中所谓的双有理截面猜想。处理这些猜想的技术包括一般赋值理论，一种研究绝对伽罗瓦群的结构，重点是它们如何"看到"赋值的方法（这已经在Koenigsmann的早期工作中得到了很好的发展），Philip Dittmann（2018）关于绝对伽罗瓦群模型理论的最新进展，来自代数数论的精炼技术，包括伽罗瓦上同调和可能来自稳定性理论的一些输入。该项目结合了三个EPSRC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.