Anabelian Geometry and Field Arithmetic II

阿纳贝尔几何与域算术 II

• 批准号: 1101397
• 负责人: Florian Pop
• 金额: $ 26.1万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101397&HistoricalAwards=false
• 关键词: Anabelian; Geometry; Field; Arithmetic; II

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.

该研究项目涉及算术和代数几何中的阿贝尔现象以及域算术中的问题的研究。 PI 计划继续开展由博戈莫洛夫发起的阿贝尔计划，该计划旨在以函子的方式从他们的亲拉贝尔中央伽罗瓦理论中恢复至少两个超越度的函数域。 PI 完成了有限域代数闭包上的函数域的程序，他计划完成全局域的代数闭包上的函数域和更一般的代数闭基域的程序。 PI 计划利用正在讨论的阿纳贝尔纲领与 Ihara/Oda-Matsumoto 猜想的关系，并使用这些方法在 Ihara/Oda-Matsumoto 猜想的几个方向上进行推广；特别是，为了证明任意基域的这个猜想。这将对理解特别是有理数域以及一般任意域的伽罗瓦结构产生重大影响。 PI（与合作者共同）也期望在格洛腾迪克（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方面取得进展。最后，PI 希望在更好地理解本地化过程（特别是哪些过程）如何导致大型字段方面取得进展。特别是，为了更好地理解本地化过程如何通过局部-全局原则与大领域和自由度猜想相关联。 PI 计划通过开发新工具以及使用先前证明并在其他环境中成功使用的一般类型结果，简化并证明关于经典伽罗瓦以及微分伽罗瓦理论背景下大域上非平凡分裂嵌入问题的可解性的更强结果。对上述问题的肯定答案将对现代伽罗瓦理论的进展以及算术中的一些非常基本的问题产生重大影响 几何和代数几何。研究结果将通过讲座和科学期刊上的出版物广泛传播到数学界。首席研究员是活动的联合组织者和高级受邀研究员，其目的是：第一，为各级国际合作、培训和科学交流创造广泛的基础；二是针对研究生和青年研究人员开展专题活动，从而增强教学和技术理解。