Anabelian Geometry and Field Arithmetic

首席研究员计划继续研究阿纳贝尔几何和场算术，以及这两门学科与一般数学的其他方面的相互作用，特别是与算术几何和代数几何的相互作用。这项研究将在一定程度上以已知的阿贝尔现象和场算术中的已知事实的可能扩展为指导，国际数学协会对此做出了重大贡献。PI期望在有限域的代数闭包、整体域的代数闭包、甚至更一般的代数闭域上证明L关于tr.次1的函数域的中心二元阿贝尔猜想。PI期望得到伊原/大田-松本猜想的按中心支持L的阿贝尔形式，这将对理解有理数领域(和其他领域)的伽罗华结构有重大影响。PI还期望在Grothendieck(p-进)部分猜想及其与(一个有效的)Mordell猜想--Faltings定理的关系上取得进展。PI期望在更好地理解大域的类以及与Freess猜想相关的域的上同调方面取得进展。对上述问题的肯定回答将对现代伽罗瓦理论的进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要的影响。结果将通过讲座和科学期刊上的出版物向数学界广泛传播。国际和平研究所是活动的共同组织者和高级特邀研究员，这些活动计划：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，为研究生和青年研究人员举办特别活动，从而增进教学和技术理解。

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.