Modular Symbols in Arithmetic

这个研究项目的重点是数论，这是一门古老的学科，起源于对整数算术性质的研究。今天，不同数学性质（代数、几何和解析）的算术对象之间的联系形成了该领域的一个标志。该奖项的研究符合这一主题，同时探索了直到最近才可能意想不到的深层联系。该研究的一般原理是，某些代数问题可以被重新解释为更高维度的几何问题：例如，某条曲线上的路径决定了某些算术上有趣的数字的“乘积”。PI的研究旨在揭示这类复杂但明确的关系，其广泛目标是提供一个窗口，通过这个窗口，人们可以从新的角度看待众所周知的算术猜想和陈述。获得该奖项的研究生将接受培训，为研究做出贡献。该奖项支持的研究涉及的是切环单位的模符号和斯坦伯格符号之间部分推测关系的类似物。这粗略地说，环场的类群是显式地由模爱森斯坦理想化后的模曲线的同调群决定并确定的：符号上的显式映射预计是由模曲线上同调上的伽罗瓦作用构造的映射的逆。该奖项的主要重点是将框架和已知结果扩展到高维代数群和其他全球领域。特别是，PI计划在原始猜想中显著推广显式映射的动机构造，并分析伽罗瓦表示，该表示应在算术兴趣的情况下给出逆映射。总的期望是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Selmer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。

The focus of this research project is in number theory, an ancient subject that grew out of the study of arithmetic properties of the integers. Today, connections between arithmetic objects of different mathematical natures (algebraic, geometric, and analytic) form a hallmark of this field. The research in this award fits into this theme while exploring deep connections that were perhaps unexpected until recently. The general principle of the research is that certain algebraic problems can be reinterpreted as geometric problems in a higher dimension: for example, paths on a certain curve determine certain “products” of arithmetically interesting numbers. The research of the PI aims to uncover intricate but explicit relationships of this sort with the broad aim of providing a window through which well-known conjectures and statements of arithmetic may be seen in a new light. Graduate students supported by the award will receive training to contribute towards the research.The research supported by this award concerns analogues of a partly conjectural relationship between modular symbols and Steinberg symbols of cyclotomic units. This says roughly that class groups of cyclotomic fields are explicitly determined by and determine homology groups of modular curves reduced modulo Eisenstein ideals: an explicit map on symbols is expected to be inverse to a map constructed from the Galois action on the cohomology of a modular curve. The primary focus of the award is the extension of the framework and known results to higher-dimensional algebraic groups and other global fields. In particular, the PI plans to significantly generalize a motivic construction of the explicit map in the original conjecture and to analyze Galois representations that should give the inverse maps in cases of arithmetic interest. The overall 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.