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Modular Symbols in Arithmetic

算术中的模符号

基本信息

批准号：
1801963
负责人：
Romyar Sharifi
金额：
$ 33万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801963&HistoricalAwards=false
关键词：
Modular Symbols Arithmetic

项目摘要

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.

代数数论的一个显著方面在于它揭示了似乎具有完全不同性质的对象之间的联系。 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，在数学的不同部分之间提供有趣的联系。通过分析性质的中介对象，已经间接地发现了大量这样的联系。 PI的研究旨在提供一个窗口，通过它可以从一个新的和更直接的角度来看待著名的算术图形和语句。PI阐明了模符号与分圆单位的杯积之间复杂而明确的关系。这一点及其提出的扩展粗略地说，分圆域的类群是明确确定的，并确定模曲线约化模爱森斯坦理想的同调群。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，为作为项目主要重点的理想配方制定必要的框架。期望的是，局部对称空间的几何应该明确地确定在伽罗瓦表示，这是说结构的塞尔默groups.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Higher Chern classes in Iwasawa theory

DOI：
10.1353/ajm.2020.0017
发表时间：
2015-12
期刊：
American Journal of Mathematics
影响因子：
1.7
作者：
F. Bleher;T. Chinburg;Richard Greenberg;M. Kakde;G. Pappas;R. Sharifi;M. Taylor
通讯作者：
F. Bleher;T. Chinburg;Richard Greenberg;M. Kakde;G. Pappas;R. Sharifi;M. Taylor

Iwasawa Theory: A Climb up the Tower

岩泽理论：爬上塔楼