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

算术几何视角下的绝对伽罗瓦群模型论

• 批准号: 2099876
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2099876
• 关键词: Model; theory; absolute; Galois; groups

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.

格罗腾迪克在他写给法尔廷斯的著名信中勾勒出了后来被称为阿贝尔几何的纲领。正是本着这个纲领的精神，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。可能会认为这个群是域上下文中基本群的类比，一旦我们引入所谓的étale拓扑，这实际上不仅仅是一个类比，通常 GK 以可计算的方式编码关于 K 的算术信息。伽罗瓦上同调的机制带来了对法尔廷斯著名定理的有效版本的希望，该定理关于亏格 >1 的代数曲线上有理点的数量有限性。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），任何其绝对伽罗瓦群与 Q 的绝对伽罗瓦群同构的域 K 都带有亨塞尔估值可整除的值群，并且具有与 Q 同构的剩余域，以及基本伽罗瓦猜想 (EGC)，即任何域 K 的绝对伽罗瓦群基本上等价于该域； Q 的 Henselian 估值具有可整除的值群，且剩余域基本等价于 Q。不难看出，EGC 隐含着 AGC，并且 AGC 等价于所谓的双有理截面猜想，这是 Koenigsmann 的一个非平凡定理。格罗滕迪克的阿贝尔几何。处理这些猜想的技术包括一般估值理论，这是对绝对伽罗瓦群结构的研究，重点是它们如何“查看”估值（Koenigsmann 的早期工作已经很好地发展了这一点）、Philip Dittmann (2018) 在绝对 Galois 群模型理论方面的最新进展、代数数论的改进技术，包括 Galois 上同调以及可能来自稳定性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。