Contents Online

# Homology, Homotopy and Applications

## Volume 7 (2005)

### Number 3

### Proceedings of a Conference in honour of Victor Snaith

### Toward equivariant Iwasawa theory, IV

Pages: 155 – 171

DOI: http://dx.doi.org/10.4310/HHA.2005.v7.n3.a8

#### Authors

#### Abstract

Let $l$ be an odd prime number and $K_{\infty}/k$ a Galois extension of totally real number fields, with $k/{\Bbb{Q}}$ and $K_{\infty}/k_{\infty}$ finite, where $k_{\infty}$ is the cyclotomic ${\Bbb{Z}}_l$-extension of $k$. In [RW2] a “main conjecture” of equivariant Iwasawa theory is formulated which for pro-$l$ groups $G_{\infty}$ is reduced in [RW3] to a property of the Iwasawa $L$-function of $K_{\infty}/k$. In this paper we extend this reduction for arbitrary $G_{\infty}$ to $l$-elementary groups $G_{\infty}=\langle s \rangle\times U$, with $\langle s \rangle$ a finite cyclic group of order prime to $l$ and $U$ a pro-$l$ group. We also give first nonabelian examples of groups $G_{\infty}$ for which the conjecture holds.

#### Keywords

Iwasawa theory, $l$-adic $L$-functions

#### 2010 Mathematics Subject Classification

11R23, 11R32, 11R37, 11R42, 11S23, 11S40