Anabelian Geometry and Field Arithmetic

首席研究员计划继续他的安娜贝尔几何和场算术的研究，以及这两个学科与数学的其他方面的相互作用，特别是算术几何和代数几何。这项研究将部分地由已知的安娜贝尔现象的可能扩展和已知的领域算术事实来指导，PI在这方面做出了重大贡献。在有限域的代数闭包上，在全局域的代数闭包上，甚至在更一般的代数闭包上，PI期望证明tr1次函数场的亲- 1阿贝尔-中心二族阿贝尔猜想。PI期望获得Ihara/Oda- Matsumoto猜想的顺阿贝尔中心形式，这将对理解有理数领域（以及其他领域）的伽罗瓦结构产生重大影响。PI还期望在Grothendieck （p-adic）截面猜想及其与（一个有效的）莫德尔猜想——法尔廷斯定理的关系上取得进展。PI期望在更好地理解大场的类别以及与自由猜想相关的场的上同性方面取得进展。对上述问题的肯定回答将对现代伽罗瓦理论的发展，以及算术几何和代数几何中的一些非常基本的问题产生非常重大的影响。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，计划做到这两点：第一，为各级国际合作，培训和科学交流创造广泛的基础，第二，为研究生和年轻研究人员举办专门的活动，从而加强教学和技术理解。

The Principal Investigator plans to continue his study of Anabelian geometry and of Field Arithmetic, as well as the interaction of these two subjects with other aspects of mathematics in general, and with arithmetic geometry and algebraic geometry in particular.This study will be guided in part by possible extensions of the known anabelian phenomena, and of the known facts from field arithmetic, to which the PI has made major contributions. The PI expects to prove the pro-l abelian-by-central birational anabelian Conjecture for function fields of tr.degree 1 over algebraic closures of finite fields, of global fields, and even more general algebraically closed fields. The PI expects to obtain the pro-l abelian-by-central form of the Ihara/Oda--Matsumoto conjecture, which would have a major impact on understanding the Galois structure of the field of rational numbers (and other fields). The PI expects as well to make progress on Grothendieck's (p-adic) Section Conjecture and its relation to (an effective) Mordell Conjecture --Faltings' Theorem. The PI expects to make progress on better understanding the class of large fields, and of the cohomology of fields as related to the Freeness Conjecture.Positive answers to the questions mentioned above would have a very 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 plan 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.