Modular Symbols in Arithmetic

代数数论的一个显著方面在于它揭示了看起来具有完全不同性质的物体之间的联系。PI研究的首要原则是，某些代数问题可以直接重新解释为更高维度的几何问题，从而在不同数学领域的对象之间提供有趣的联系。大量的这种联系是通过分析性质的中间对象间接地发现的。PI的研究旨在提供一个窗口，通过这个窗口，人们可以从新的和更直接的角度来看待众所周知的算术猜想和陈述。PI推测了模符号和切环单位的杯积之间复杂而明确的关系。这一结论及其推广表明，切环场的类群是由简化模爱森斯坦理想的模曲线的同调群显式确定并确定的。该项目旨在加强他的猜想，并将其和已知结果扩展到高维代数群和其他全球领域。这涉及大量的基础工作，为项目的主要焦点所期望的配方开发必要的框架。期望局部对称空间的几何应该明确地决定伽罗瓦表示中的格的算术，也就是塞尔默群的结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A remarkable aspect of algebraic number theory lies in its revealing of connections between objects that appear to be of entirely different natures. The overarching principle of the research of the PI is that certain algebraic problems can be reinterpreted directly as geometric problems in higher dimension, providing intriguing connections between objects from different parts of mathematics. A great wealth of such connections have been found indirectly through intermediate objects of an analytic nature. The research of the PI aims to provide a window through which well-known conjectures and statements of arithmetic may be seen in a new and more direct light.The PI has conjectured an intricate but explicit relationship between modular symbols and cup products of cyclotomic units. This and its proposed extensions say roughly that class groups of cyclotomic fields are explicitly determined by and determine homology groups of modular curves reduced modulo Eisenstein ideals. The project aims to strengthen his conjecture and extend it and known results to higher-dimensional algebraic groups and other global fields. This involves substantial foundational work to develop the necessary framework for the desired formulations that are the primary focus of the project. The expectation is that the geometry of locally symmetric spaces should explicitly determine the arithmetic of lattices in Galois representations, which is to say the structure of Selmer groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.