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Modular Symbols in Arithmetic

算术中的模符号

基本信息

批准号：
2101889
负责人：
Romyar Sharifi
金额：
$ 37万
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：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101889&HistoricalAwards=false
关键词：
Modular Symbols Arithmetic

项目摘要

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.

这个研究项目的重点是数论，这是一个古老的学科，起源于整数的算术性质的研究。今天，不同数学性质（代数、几何和分析）的算术对象之间的联系形成了这个领域的标志。该奖项的研究符合这一主题，同时探索了直到最近才意想不到的深层联系。研究的一般原则是，某些代数问题可以重新解释为更高维度的几何问题：例如，某条曲线上的路径确定算术有趣数的某些“乘积”。PI的研究旨在揭示此类复杂但明确的关系，其广泛目标是提供一个窗口，通过该窗口可以从新的角度看待众所周知的猜想和算术陈述。该奖项支持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分圆单位的模符号和Steinberg符号之间的部分几何关系的类似物。这大概是说，分圆域的类群明确地由模曲线的模艾森斯坦理想约化的同调群确定，并且确定模曲线的模艾森斯坦理想约化的同调群：符号上的显式映射被期望与由模曲线的上同调上的伽罗瓦作用所构造的映射是逆的。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计划显着推广原猜想中显式映射的motivic构造，并分析在算术兴趣的情况下应该给出逆映射的伽罗瓦表示。总体的期望是，局部对称空间的几何应该明确地确定伽罗瓦表示中的格的算术，也就是说，塞尔默群的结构。这个奖项反映了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。