Arithmetic Geometry: Topics in Iwasawa Theory

算术几何：岩泽理论专题

基本信息

批准号：
1801328
负责人：
Frauke Bleher
金额：
$ 19万
依托单位：
University of Iowa
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801328&HistoricalAwards=false
关键词：
Arithmetic Geometry Topics Iwasawa Theory

项目摘要

A common theme in science is the study and measurement of growth rates. This research project investigates growth rates of arithmetic invariants related to the symmetry groups of algebraic equations in infinite families. This work was begun in the 1950's by Kenkichi Iwasawa, who conjectured that many such families have a well-defined asymptotic behavior. The proofs of such "main conjectures" have been at the forefront of algebraic number theory. To date, research has focused on first-order estimates of growth rates. The goal of this project is to study the higher-order terms, beyond their first-order terms. This will lead to more precise measurements of the growth rates in question. The project's broader significance is that it will provide a better understanding of important methods in algebraic number theory that have in the past led to the development of technology essential to society, such as the improved compression and secure transmission of data.Iwasawa theory has played an essential role in the development of modern number theory and arithmetic geometry. This project concerns a new frontier in the subject, namely the study of the higher-order terms of Iwasawa and Selmer modules. One focus will be on conjectures about when the first-order terms are zero and what one should expect for the leading terms in such cases. A main goal is to formulate a general framework for studying higher-order terms in Iwasawa theory using a two-step approach: The first step is to define a "large" Selmer module via the Galois cohomology of a motive or a family of Galois representations; the second step is to impose local conditions on the cohomology classes to produce a "small" Selmer module. Greenberg formulated a "main conjecture" in this generality, which can be thought of as concerning the first-order term of the small Selmer module. The overall goal of this project is to develop a higher-order term Iwawasa theory in the same generality and to explore its consequences.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.

科学的一个共同主题是增长率的研究和测量。该研究项目研究与无限族中代数方程对称群相关的算术不变量的增长率。这项工作由 Kenkichi Iwasawa 于 1950 年代开始，他推测许多这样的家庭都有明确定义的渐近行为。这种“主要猜想”的证明一直处于代数数论的前沿。迄今为止，研究主要集中在增长率的一阶估计上。该项目的目标是研究一阶项之外的高阶项。这将导致对所讨论的增长率进行更精确的测量。该项目更广泛的意义在于，它将提供对代数数论中的重要方法的更好理解，这些方法在过去导致了对社会至关重要的技术的发展，例如改进的数据压缩和安全传输。岩泽理论发挥了重要作用在现代数论和算术几何的发展中发挥着至关重要的作用。该项目涉及该学科的一个新领域，即 Iwasawa 和 Selmer 模的高阶项的研究。一个重点是关于一阶项何时为零的猜想以及在这种情况下对领先项的期望。主要目标是使用两步方法制定研究岩泽理论中高阶项的通用框架：第一步是通过动机或伽罗瓦表示族的伽罗瓦上同调来定义“大”Selmer 模块;第二步是对上同调类施加局部条件以产生“小”Selmer 模块。格林伯格在这个一般性中提出了一个“主要猜想”，它可以被认为是关于小塞尔默模的一阶项。该项目的总体目标是开发具有相同通用性的高阶术语 Iwawasa 理论并探索其后果。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Exterior powers in Iwasawa theory

岩泽理论中的外部权力

DOI：
10.4171/jems/1115
发表时间：
2022-01
期刊：
Journal of the European Mathematical Society
影响因子：
2.6
作者：
Bleher, Frauke;Chinburg, Ted;Greenberg, Ralph;Kakde, Mahesh;Sharifi, Romyar;Taylor, Martin J.
通讯作者：
Taylor, Martin J.

Galois structure of the holomorphic differentials of curves

曲线全纯微分的伽罗瓦结构

DOI：
10.1016/j.jnt.2020.04.015
发表时间：
2020-11
期刊：
Journal of Number Theory
影响因子：
0.7
作者：
Bleher, Frauke M.;Chinburg, Ted;Kontogeorgis, Aristides
通讯作者：
Kontogeorgis, Aristides

Higher Chern classes in Iwasawa theory

岩泽理论的陈省身高级课程

DOI：
10.1353/ajm.2020.0017
发表时间：
2020-01
期刊：
American Journal of Mathematics
影响因子：
1.7
作者：
Bleher, F. M.;Chinburg, T.;Greenberg, R.;Kakde, M.;Pappas, G.;Sharifi, R.;Taylor, M. J.
通讯作者：
Taylor, M. J.

On representations of Gal(Q‾/Q) , GTˆ and Aut(Fˆ2)

关于 Gal(Q−/Q) 、GT− 和 Aut(F−2) 的表示