In this paper we investigate reduction of nontorsion elements in the étale K-theory of a curve X over a global field F. We show that the reduction map can be well understood in terms of Galois cohomology of l-adic representations.

We describe a criterion for a natural Euler characteristic that takes values in a relative algebraic K ₀-group to be additive in distinguished triangles. As preliminary steps we prove several results about determinant functors, in particular concerning the comparison of the determinant of a complex to the determinant of its cohomology.

We compute the entire cyclic homology and cohomology of the Schatten ideals fl ^p .

We recall coherent definitions of two commutative local reciprocity homomorphisms, arithmetic and geometric, and then suggest a new approach to the description of the image of a noncommutative local reciprocity map introduced in [2] and discuss some of its properties in relation to the commutative maps.

We survey available results on the construction and calculation of equivariant versions of real and complex connective K-theory.

We prove an equivariant Grothendieck-Ogg-Shafarevich formula. This formula may be viewed as an étale analogue of well-known formulas for Zariski sheaves generalizing the classical Chevalley-Weil formula. We give a new approach to those formulas (first proved by Ellingsrud/Lønsted, Nakajima, Kani and Ksir) which can also be applied in the étale case.

Explicit Brauer Induction formulae with certain natural behaviour have been developed for complex representations, for example by work of Boltje, Snaith and Symonds. In this paper we present induction formulae for symplectic and orthogonal representations of finite groups. The problems are motivated by number theoretical and topological questions. We will prove naturality with respect to restriction and inflation. Also we investigate complexification maps and use them to compare the orthogonal and symplectic induction formulae with Boltje's complex induction formula.

Let l be an odd prime number and K _$infin$/k a Galois extension of totally real number fields, with k/Q and K _$infin;/k _$infin; finite, where k _$infin$ is the cyclotomic Z _l -extension of k. In [RW2] a "main conjecture" of equivariant Iwasawa theory is formulated which for pro-l groups G _$infin$ is reduced in [RW3] to a property of the Iwasawa L-function of K _$infin$/k. In this paper we extend this reduction for arbitrary G _$infin$ to l-elementary groups G _$infin$=$lang$s$rang$ x U, with $lang$s$rang$ a finite cyclic group of order prime to l and U a pro-l group. We also give first nonabelian examples of groups G _$infin$ for which the conjecture holds.

This paper is a survey on recent researches of the author and his recent joint work with Shuji Saito. We will explain how to construct p-adic étale Tate twists on regular arithmetic schemes with semistable reduction, and state some fundamental properties of those objects. We will also explain how to define cycle class maps from Chow groups to étale cohomology groups with coefficients in p-adic étale Tate twists and state injectivity and surjectivity results on those new cycle class maps.