Anabelian methods in arithmetic and algebraic geometry

算术和代数几何中的阿纳贝尔方法

• 批准号: RGPIN-2022-03116
• 负责人: Litt, Daniel
• 金额: $ 1.89万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758686
• 关键词: Anabelian; methods; arithmetic; algebraic; geometry

This proposal aims to further develop the beautiful, though not yet well-understood, connections between algebraic geometry, number theory, and topology. Algebraic varieties -- the sets of solutions to systems of polynomial equations -- are ubiquitous in mathematics, and their fundamental nature comes in part from the fact that their study lies at the intersection of such varied fields of mathematics. The goal of this proposal is to unwind the connections between these seemingly disparate fields, primarily through the study of the fundamental group of an algebraic variety, an invariant which loosely speaking captures the structure of loops in the variety. Recent discoveries have shown that this invariant is crucial to understanding, for example, rational solutions to systems of polynomial equations -- such so-called "Diophantine" questions have fascinated mathematicians for millenia. Despite their long history, we are only now beginning to understand the connections of such questions to topology, via the section conjecture, the non-abelian Chabauty method, and other extremely recent developments. Broadly speaking, the aspect of algebraic geometry and number theory connected to the fundamental group is called "anabelian geometry," which is the subject of the proposal. Building on my previous work, I plan to better understand anabelian aspects of the topology of algebraic varieties, and in particular the relationship between anabelian geometry and monodromy representations, with the goal of proving two well-known open questions in geometry: the geometric torsion conjecture and the (conjectural) Hard Lefschetz theorem in positive characteristic. Progress on these questions would fundamentally advance our understanding of the topology of algebraic varieties. I also plan to make progress (in joint work with Aaron Landesman) on the Putman-Wieland conjecture, a fundamental question in the topology of surfaces, by bringing to bear algebro-geometric and topological techniques; similarly, my joint work with Li, Salter, and Srinivasan shows that such techniques can yield insight into Grothendieck's section conjecture, perhaps the fundamental (conjectural) connection between anabelian geometry and arithmetic. This work will also yield insight into the topology of moduli spaces, one of the fundamental objects of study in algebraic geometry. Finally, this proposal will build on very recent developments in arithmetic geometry -- in particular, the non-abelian Chabauty method -- to develop practical methods for solving arithmetic questions. In particular, joint work with Eric Katz will yield techniques for running the non-abelian Chabauty method to find rational points on curves of bad reduction, which will be crucial to make the method practical as a way to find solutions to systems of polynomial equations.

该提案旨在进一步发展代数几何，数字理论和拓扑之间的美丽，尚不理解的美丽，尽管尚未得到充分理解。代数品种 - 多项式方程系统的解决方案集合在数学上是无处不在的，它们的基本性质部分源于它们的研究在于这种数学多样性领域的交集。该提案的目的是放松这些看似不同的领域之间的联系，主要是通过对代数品种的基本群体的研究，这种群体的不变型逐渐捕获了多种多样的循环结构。最近的发现表明，这种不变对于理解，例如，对多项式方程系统的理性解决方案至关重要 - 这样的“二只”问题使数学家对Millenia着迷。尽管历史悠久，但我们直到现在才开始通过“分节协议”，“非亚洲chabauty方法”以及其他最新的发展来理解此类问题与拓扑的联系。从广义上讲，与基本群体相关的代数几何形状和数字理论的方面称为“ anabelian几何形状”，这是该提议的主题。在我以前的工作的基础上，我计划更好地理解代数品种拓扑的阿纳贝利亚方面，尤其是阿纳贝尔几何形状和单片表示之间的关系，目的是证明几何学中的两个众所周知的开放性问题：几何概念和（猜测）硬Lefschetz hard lefschetz theorem inorem inorem in of acturentiment。这些问题的进展从根本上可以提高我们对代数品种拓扑的理解。我还计划在Putman-Wieland Concept上（与Aaron Landesman的联合合作），这是表面拓扑的一个基本问题，通过带来代数几何和拓扑技术；同样，我与Li，Salter和Srinivasan的联合合作表明，这种技术可以洞悉Grothendieck的部分会议，也许是Anabelian几何形状和算术之间的基本（猜想）联系。这项工作还将深入了解莫德利空间的拓扑，这是代数几何学研究的基本对象之一。最后，该提案将基于算术几何形状的最新发展，特别是非亚洲chabauty方法，以开发解决算术问题的实用方法。特别是，与埃里克·卡兹（Eric Katz）的联合合作将产生运行非亚洲chabauty方法的技术，以在不良降低的曲线上找到理性点，这对于使该方法的实用性作为找到多项式方程系统解决方案的一种方式至关重要。