Multirelative -theory and axioms for the -theory of rings.
Natural homomorphisms of Witt rings of orders in algebraic number fields. II
We prove that there are infinitely many real quadratic number fields with the property that for infinitely many orders in and for the maximal order in the natural homomorphism of Witt rings is surjective.
Natural homomorphisms of Witt rings of orders in algebraic number fields
Non-abelian congruences between -values of elliptic curves
Let be a semistable elliptic curve over . We prove weak forms of Kato’s -congruences for the special values More precisely, we show that they are true modulo , rather than modulo . Whilst not quite enough to establish that there is a non-abelian -function living in , they do provide strong evidence towards the existence of such an analytic object. For example, if these verify the numerical congruences found by Tim and Vladimir Dokchitser.
Non-commutative differential geometry
Noncommutative localisation in algebraic -theory. I.
Noncommutative numerical motives, Tannakian structures, and motivic Galois groups
In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric...
On analytic torsion over C*-algebras
In this paper, we present an analytic definition for the relative torsion for flat C*-algebra bundles over a compact manifold. The advantage of such a relative torsion is that it is defined without any hypotheses on the flat C*-algebra bundle. In the case where the flat C*-algebra bundle is of determinant class, we relate it easily to the L^2 torsion as defined in [7],[5].
On Bott-periodic algebraic K-theory.
Let K*(A;Z/ln) denote the mod-ln algebraic K-theory of a Z[1/l]-algebra A. Snaith ([14], [15], [16]) has studied Bott-periodic algebraic theory Ki(A;Z/ln)[1/βn], a localized version of K*(A;Z/ln) obtained by inverting a Bott element βn. For l an odd prime, Snaith has given a description of K*(A;Z/ln)[1/βn] using Adams maps between Moore spectra. These constructions are interesting, in particular for their connections with Lichtenbaum-Quillen conjecture [16].In this paper we obtain a description...
On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields
A large number of papers have contributed to determining the structure of the tame kernel of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]). We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic forms,...
On existing of filtered multiplicative bases in group algebras.
On extensions of mixed motives.
In this article we give an introduction to mixed motives and sketch a couple of ways to construct examples.
On finite subgroups of groups of type VF.
On Galois descent for Hochschild and cyclic homology.
On generalized homology of Artin groups.
On Hochschild and cyclic homology of certain homogeneous spaces
On irreducible projective representations of finite groups.
On Jannsen's conjecture for Hecke characters of imaginary quadratic fields.
We present a collection of results on a conjecture of Jannsen about the p-adic realizations associated to Hecke characters over an imaginary quadratic field K of class number 1.The conjecture is easy to check for Galois groups purely of local type (Section 1). In Section 2 we define the p-adic realizations associated to Hecke characters over K. We prove the conjecture under a geometric regularity condition for the imaginary quadratic field K at p, which is related to the property that a global Galois...
On modified Reedy and modified projective model structures.
On nonsingular matrices and Bott periodicity.