Arithmetic Geometry: Topics in Iwasawa Theory

科学中的一个共同主题是研究和测量增长率。本研究项目研究无限族中与代数方程对称群有关的算术不变量的增长率。这项工作由岩泽健一在20世纪50年代的S开始，他推测许多这样的家庭都有一个明确的渐近行为。这种“主要猜想”的证明一直处于代数论的前沿。到目前为止，研究的重点是增长率的一阶估计。这个项目的目标是研究高阶项，超越它们的一阶项。这将导致对所讨论的增长率进行更准确的测量。该项目更广泛的意义在于，它将更好地理解代数论中的重要方法，这些方法在过去导致了对社会至关重要的技术的发展，如改进的数据压缩和安全传输。岩泽理论在现代数论和算术几何的发展中发挥了重要作用。本项目涉及该学科的一个新的前沿，即岩泽模和Selmer模的高阶项的研究。其中一个焦点将是关于一阶项何时为零的猜测，以及在这种情况下人们应该对主导项期待什么。一个主要的目标是用两步法建立一个研究岩泽理论中高阶项的一般框架：第一步是通过一个动机或伽罗瓦表示族的伽罗华上同调定义一个“大的”塞尔默模；第二步是对上同调类施加局部条件以产生一个“小的”塞尔默模。在这个概括性中，格林伯格提出了一个“主要猜想”，它可以被认为是关于小塞尔默模的一阶项。这个项目的总体目标是在相同的概括性中发展更高阶岩河理论并探索其结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.