arXiv subjects
Publications and source records attributed to Jürgen Ritter.
We extend the main result of [Math. Res. Lett. 15 (2008), 715-725] to Galois extensions L/K of totally real number fields of arbitrary odd prime power degree, thereby offering support for the validity of the 'main conjecture' of equivariant Iwasawa theory.
Assuming that Iwasawa's $μ_{K/k}$-invariant vanishes, we prove the 'main conjecture' of equivariant Iwasawa theory, at odd prime numbers $l$, for arbitrary extensions $K/k$ of totally real number fields, up to its uniqueness assertion.
The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian $l$-adic Artin $L$-functions over totally real number fields. In order to generalize to non-abelian $l$-adic $L$-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the ``equivariant main conjecture'' of Iwasawa theory is discussed.
Following Deligne and Ribet (`Values of abelian $L$-functions at negative integers over totally real fields.' Invent. Math. 59 (1980), 227-286) we prove that the `torsion congruences' (as introduced in our paper `Non-abelian pseudomeasures and congruences between abelian Iwasawa $L$-functions.' To appear in Pure and Applied Mathematics Quarterly) hold and so reduce the `main conjecture' of equivariant Iwasawa theory to the integrality of the logarithmic pseudomeasure.
Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We determine the unit group of $Λ_\wedge[H]$ (the completion of the localization at $l$ of $\Bbb{Z}_l[[T]][H]$) as well as the kernel and cokernel of the integral logarithm $L:Λ_\wedge[H]^\times\to Λ_\wedge[H]$, which appears in non-commutative Iwasawa theory.
The equivariant `main conjecture' of Iwasawa theory is shown to hold for a Galois extension $K/k$ of number fields with Galois group an $l$-adic pro-$l$ Lie group of dimension 1 containing an abelian subgroup of index $l$, provided that Iwasawa's $μ$-invariant $μ(K/k)$ vanishes.