Iwasawa theory for Artin representations I

Iwasawa Theory for Elliptic Curves

The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of ZZp-extensions in the early 1970s. We first recall some results from Iwasawa’s theory. Suppose that F is a finite extension of Q and that F∞ is a Galois extension of F such that Gal(F∞/F ) ∼= ZZp, the additive group of p-adic integers, where p is any prime. Equivalentl...

Self-dual Artin Representations

Functional equations in number theory are relations between an L-function and some sort of dual L-function, and in general, the L-function and its dual need not coincide. For example, if χ is a primitive Dirichlet character then the functional equation relates L(s, χ) to L(1− s, χ), and L(s, χ) = L(s, χ) if and only if χ = 1. Or if f is a primitive cusp form of weight two for Γ1(N) and f∨ is th...

Introduction to Iwasawa Theory

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.

Refined Iwasawa theory for p-adic representations and the structure of Selmer groups

In this paper, we develop the idea in [16] to obtain finer results on the structure of Selmer modules for p-adic representations than the usual main conjecture in Iwasawa theory. We determine the higher Fitting ideals of the Selmer modules under several assumptions. Especially, we describe the structure of the classical Selmer group of an elliptic curve over Q, using the ideals defined from mod...