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、数论和拓扑之间美丽但尚未充分理解的联系。代数簇（多项式方程组的解集）在数学中无处不在，而且它们的基本原理也是如此。本质部分来自于这样一个事实：他们的研究位于这些不同数学领域的交叉点。该提案的目标是主要通过对基本组的研究来解开这些看似不同领域之间的联系。代数簇，一种不变量，从广义上讲，它捕获了簇中循环的结构，最近的发现表明，这种不变量对于理解多项式方程组的有理解（例如所谓的“丢番图”问题）至关重要。尽管它们的历史悠久，但我们现在才开始通过截面猜想、非阿贝尔查博蒂方法和其他方法来理解这些问题与拓扑的联系。从广义上讲，与基本群相关的代数几何和数论方面被称为“阿纳贝尔几何”，这是该提案的主题，基于我之前的工作，我计划更好地理解该学科的阿纳贝尔方面。代数簇的拓扑，特别是阿贝尔几何和单数表示之间的关系，目的是证明几何中两个众所周知的开放问题：几何扭转猜想和（猜想的）硬Lefschetz 定理的积极特征。这些问题的进展将从根本上增进我们对代数簇拓扑的理解。我还计划（与 Aaron Landesman 合作）在 Putman-Wieland 猜想上取得进展，这是拓扑中的一个基本问题。类似地，我与 Li、Salter 和 Srinivasan 的合作表明，这些技术可以深入了解格洛腾迪克的部分。猜想，也许是阿贝尔几何和算术之间的基本（猜想）联系这项工作还将深入了解模空间的拓扑，这是代数几何的基本研究对象之一。算术几何——特别是非阿贝尔查鲍蒂方法——开发解决算术问题的实用方法。特别是，与埃里克·卡茨的联合工作将产生运行非阿贝尔数学的技术。 Chabauty 方法在不良约简曲线上找到有理点，这对于使该方法作为求解多项式方程组的方法实用化至关重要。