The aim of this expository article is to discuss the μ-invariant associated to finitely generated modules over Iwasawa algebras. This is an important invariant which was first discovered by Iwasawa in the 1960’s and occurs in his seminal work on cyclotomic fields [9], [10]. In fact, Iwasawa conjectured that his μ-invariant was always zero for the pprimary subgroup of the ideal class group of th...

We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture.

We discuss abelian equivariant Iwasawa theory for elliptic curves over $${\mathbb {Q}}$$ at good supersingular primes and non-anomalous ordinary primes. Using Kobayashi’s method, we construct Coleman maps, which send the Beilinson–Kato element to p-adic L-functions. Then propose main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application...

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...

(typeset by Robert JS McDonald, UConn1) Lecture 1. Foundational material. The lecture will briefly cover, without proofs, the background in algebra and number theory needed at the beginning of Iwasawa theory. Throughout, p will denote an arbitrary prime number, and Γ a topological group which is isomorphic to the additive group of p-adic integers Zp. Thus, for each n ≥ 0, Γ will have a closed s...

In these notes, we follow the proof in [1] of the main conjecture of Iwasawa theory making heavy use of the Euler system of cyclotomic units. On the one hand, using the local theory of Coleman series and ideas of Iwasawa one obtains a connection with the p-adic zeta function. On the other hand by CFT and Rubin’s refinement of the ideas of Kolyvagin (and the analytic class number formula) one ob...