Toward equivariant Iwasawa theory, IV

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

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Published 1 January 2005