We derive a power series formula for the p-adic regulator on the higher dimensional algebraic K-groups of number fields. This formula is designed to be well suited to computer calculations and to reduction modulo powers of p.

The fractional Galois ideal is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higher $K$-groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Stark's conjectures and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras.

Let l be an odd prime and let A be a commutative ring containing 1/l. Let K*(A;Z/lv) denote the mod lv algebraic K-theory of A [3]. As explained in [4] there exists a “Bott element” βv∈K21v–1(l–1)(Z[1/l];Z/lv) and, using the K-theory product we may, following [16, Part IV], form

which is defined as the direct limit of iterated multiplication by βv. There is a canonical localisation map

We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraic $K$-groups of rings of integers in number fields. Our conjecture ismotivic in the sense that it involves the (transcendental) Borel regulator as well as being related to $l$–adic étale cohomology. In addition, the conjecture generalises the wellknown Coates–Sinnott conjecture. For example, for a totally real extension when $r\,=\,-2,\,-4,\,-6,\,\ldots $ the Coates–Sinnott conjecture merely predicts that zero annihilates ${{K}_{-2r}}$ of the ring of $S$–integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the étale cohomology of the cyclotomic extensions of the rationals.

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which $G$ is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures of Coates and Sinnott, and Brumer. Higher Stickelberger ideals are defined in terms of special values of L-functions; when these vanish we show how to define fractional ideals, generalising the Stickelberger ideals, with similar annihilator properties. The fractional ideal is constructed from the Borel regulator and the leading term in the Taylor series for the L-function. En route, our methods yield new proofs, in the case of abelian number fields, of formulae predicted by Lichtenbaum for the orders of K-groups and étale cohomology groups of rings of algebraic integers.

Inspired by the work of Bloch and Kato in [2], David Burns constructed several ‘equivariant Tamagawa invariants’ associated to motives of number fields. These invariants lie in relative $K$-groups of group-rings of Galois groups, and in [3] Burns gave several conjectures (see Conjecture 3.1) about their values. In this paper I shall verify Burns' conjecture concerning the invariant $T\Omega^{\rm loc}( N/{\bf Q},1)$ for some families of quaternion extensions $N/{\bf Q}$. Using the results of [9] I intend in a subsequent paper to verify Burns' conjecture for those families of quaternion fields which are not covered here.

In this paper BP-theory is used to give a proof that there exists a stable homotopy element in \pi _{2^{n+1} - 2}^{S}( {\tf="times-b"R}P^{\infty }) with non-zero Hurewicz image in ju-theory if and only if there exists an element of \pi _{2^{n+1} - 2}^{S}( S{\hskip1}^{0}) that is represented by a framed manifold of Arf invariant one.

An alternative proof is given of a result, originally due to Guido Mislin, giving necessary and sufficient conditions for the inclusion of a subgroup to induce an isomorphism in mod p cohomology.