Arithmetic Geometry: Topics in Iwasawa Theory

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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.