Arithmetic statistics and noncommutative Iwasawa theory

Arithmetic incarnations of zeta in Iwasawa theory

We give a brief exposition of the Iwasawa theory of cyclotomic extensions, so as to discuss its relationship with p-adic zeta functions. We give an overview of these connections and the arithmetic significance of the theory, leading up to a statement of the main conjecture.

Algorithms for the arithmetic of elliptic curves using Iwasawa theory

We explain how to use results from Iwasawa theory to obtain information about p-parts of Tate-Shafarevich groups of specific elliptic curves over Q. Our method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is g...

Noncommutative Geometry and Arithmetic

This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk concentrates on two main aspects: the relation of Stark numbers to the geometry of noncommutative...

Iwasawa theory and generalizations

This is an introduction to Iwasawa theory and its generalizations. We discuss some main conjectures and related subjects. Mathematics Subject Classification (2000). Primary 11R23; Secondary 11G40, 14G10.

Introduction to Iwasawa Theory