Jürgen Ritter

Torsion units in integral group rings of some metabelian groups, II

Abstract We prove that any torsion unit of the integral group ring Z G is rationally conjugate to a trivial unit if G = A ⋊ X with both A and X abelian, | Xz . sfnc ; p for every prime p dividing | A | provided either | X | is prime or A ic cyclic.

Toward equivariant Iwasawa theory, II

Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G∞, X∞ the Galois groups of K∞/k and M∞/K∞, respectively, where K∞ is the cyclotomic l-extension of K and M∞ the maximal abelian S-ramified l-extension of K∞ with S a sufficiently large finite set of primes of k. We introduce a new K-theoretic variant of the Iwasawa ℤ[[G∞]]-module X∞ and, for K/k abelian, formulate a conjecture, which is the main conjecture of classical Iwasawa theory when ll[K : k]. We prove this new conjecture when Iwasawas μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l.

Integral group rings with trivial central units

Let It = %{ZG) be the unit group of the integral group ring ZC. It is a classical result of G. Higman (Sehgal, 1978, p. 57) that if G is torsion then IL is trivial if and only if G is abelian of exponent 2, 3, 4 or 6 or G = ^s x E, the product of the quaternion group K^ of order 8 and an elementary abelian 2-group E. Since triviality of IL for torsion free groups G is an open problem it is quite difficult to extend this result to arbitrary groups. However, if a mild condition is imposed on G a classification can be seen from the results in Sehgal (1978) as we show in Sec. 5. In 1990, Ritter and Sehgal classified finite groups G so that the central units of ZG are trivial, namely of the form ±g, where g is an element of the centre of G. A classification of arbitrary groups G with trivial central units in ZG was asked for in Sehgal (1993, Problem 26, p. 301). Recently, Parmenter (1999) renewed this call. We give below this classification in terms of the condition of Ritter and Sehgal (1990).

Cohomology of units and -values at zero

Let Klk be a finite Galois extension of number fields with Galois group G, and let S be a finite G-stable set of primes of K containing all archimedean primes. We denote the G-module of S-units of K by E = Es and let AS be the kernel of the augmentation map ZS -* Z which sends each basis element p C S to 1. We are concerned with invariants of K/k which are associated to a G-homomorphism p: AS -* E inducingQ0 AS cQ0E. These invariants were defined by Tate [Ta2] and Chinburg [Chl] when S is large, i.e. when S contains all ramified primes of K/k and the S-class group cl = cls of K is trivial. There are two of them and each is a function of the complex characters X of G. The first, AW , has A. (X) equal to the Tate regulator at X divided by the leading coefficient c(x) of the Taylor expansion at s = 0 of the Artin L-function L(s, X) with the Euler factors at primes of S omitted. According to Starks conjecture [St] in Tates form, A., takes algebraic values which satisfy A

The lifted root number conjecture and Iwasawa theory

Introduction The Tripod Restriction, deflation change of maps, and variance with

Constructible units in abelian p-group rings

S

REPRESENTING Ω(l∞) FOR REAL ABELIAN FIELDS

Definition of

L-values at Zero and the Galois Structure of Global Units

\mho_S

The frattini subgroup of the absolute Galois group of a local field

\Omega_\Phi