Anabelian Geometry and Field Arithmetic II

本研究项目涉及算术和代数几何中的可倒性现象的研究，以及野外算术中的问题。PI计划继续他在由Bogomolov发起的一个anabelian计划上的工作，该计划旨在以泛函的方式从他们的亲阿贝尔-中心伽罗瓦理论中恢复至少两个超越度的函数场。PI完成了有限域代数闭包上的函数场的程序，他计划完成全局域代数闭包上的函数场和更一般的代数闭基域的程序。PI计划利用所讨论的anabelian规划与Ihara/Oda-Matsumoto猜想的关系，并利用这些方法给出Ihara/Oda-Matsumoto猜想在几个方向上的推广；特别地，在任意基域下证明了这个猜想。这将对理解有理数域的伽罗瓦结构，特别是对任意域的伽罗瓦结构，产生重大影响。PI（与合作者）也期望在Grothendieck的（p-adic）截面猜想及其与（有效的）莫德尔猜想——法尔廷斯定理的关系上取得进展。最后，PI希望在更好地理解本地化过程（特别是哪些过程）如何导致大领域方面取得进展。特别是，为了更好地理解本地化过程如何通过局部-全局原则与大域和自由猜想相关联。PI计划通过开发新工具和使用先前证明并在其他情况下成功使用的一般类型的结果，简化和证明有关经典伽罗瓦和微分伽罗瓦理论背景下大域上非平凡分裂嵌入问题可解性的更强的结果。对上述问题的积极回答将对现代伽罗瓦理论的进步以及算术几何和代数几何中的一些非常基本的问题产生重大影响。研究结果将通过科学期刊上的讲座和出版物广泛传播给数学界。PI是活动的共同组织者和高级特邀研究员，其目的是：首先，为各级的国际合作、培训和科学交流创造广泛的基础；第二，为研究生和青年研究人员举办专门的活动，从而加强教学和技术理解。

This research project concerns the study of anabelian phenomena in arithmetic and algebraic geometry as well as questions in field arithmetic. The PI plans to continue his work on an anabelian program initiated by Bogomolov, which aims at recovering function fields of transcendence degree at least two from their pro-l abelian-by-central Galois theory in a functorial way. The PI completed that program for function fields over algebraic closures of finite fields, and he plans to complete that program for function fields over algebraic closures of global fields and more general algebraically closed base fields. The PI plans to exploit the relation of the anabelian program under discussion with the Ihara/Oda-Matsumoto conjecture, and to use these methods to give generalizations in several directions of the Ihara/Oda-Matsumoto conjecture; in particular, to prove this conjecture for arbitrary base fields. This would have a major impact on understanding the Galois structure of the field of rational numbers in particular, and of arbitrary fields in general. The PI (jointly with collaborators) expects as well to make progress on Grothendieck's (p-adic) section conjecture and its relation to (an effective) Mordell conjecture --Faltings' Theorem. Finally, the PI expects to make progress on better understanding how localization processes --in particular, which such processes-- lead to large fields. In particular, to gain a better understanding of how localization processes relate via local-global principles to large fields and the Freeness Conjecture. The PI plans to simplify and prove stronger results concerning the solvability of non-trivial split embedding problems over large fields in classical Galois, as well as differential Galois, theoretical context, both by developing new tools and by using results of general type proved previously and used successfully in other context.Positive answers to the questions mentioned above would have a significant impact on the progress of modern Galois theory and on some of the very fundamental questions in arithmetic geometry and algebraic geometry. The results will be widely disseminated to the mathematical community via talks and publications in scientific journals. The PI is co-organizer of, and senior invited researcher at, activities which aim to do both: first, to create a broad basis for international cooperation, training, and scientific exchange at all levels; and second, to have special activities for graduate students and young researchers, thus enhancing teaching and technological understanding.