Wild Kernels and divisibility in K-groups of global fields

In this paper we study the divisibility and the wild kernels in algebraic K-theory of global fields $F.$ We extend the notion of the wild kernel to all K-groups of global fields and prove that Quillen-Lichtenbaum conjecture for $F$ is equivalent to the equality of wild kernels with corresponding groups of divisible elements in K-groups of $F.$ We show that there exist generalized Moore exact sequences for even K-groups of global fields. Without appealing to the Quillen-Lichtenbaum conjecture we show that the group of divisible elements is isomorphic to the corresponding group of \' etale divisible elements and we apply this result for the proof of the $lim^1$ analogue of Quillen-Lichtenbaum conjecture. We also apply this isomorphism to investigate: the imbedding obstructions in homology of $GL,$ the splitting obstructions for the Quillen localization sequence, the order of the group of divisible elements via special values of $\zeta_{F}(s).$ Using the motivic cohomology results due to Bloch, Friedlander, Levine, Lichtenbaum, Morel, Rost, Suslin, Voevodsky and Weibel, which established the Quillen-Lichtenbaum conjecture, we conclude that wild kernels are equal to corresponding groups of divisible elementsComment: 36 page

Hecke characters and the KK-theory of totally real and CM number fields

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for $F$ with infinite type equal to a special value of certain $G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l> 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberger-splitting maps in the $l$-primary Quillen localization sequence for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$. We also apply the Stickelberger-splitting maps to construct special elements in the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$, for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general totally real number field, we study the cyclicity of $D(n)_l$ in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if $F$ is CM, special values of our $l$-adic Hecke characters are used to construct Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler system of Gauss sums and of the $K$-theoretic Euler systems constructed in Banaszak-Gajda when $K = \Bbb Q$.Comment: 38 page

Motivic Serre group, algebraic Sato-Tate group and Sato-Tate conjecture

We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under which verification of the Sato-Tate conjecture reduces to the identity connected component of the corresponding Sato-Tate group.Comment: 34 pages; restriction to odd weight adde

Detecting linear dependence by reduction maps

We consider the local to global principle for detecting linear dependence of points in groups of the Mordell-Weil type. As applications of our general setting we obtain corresponding statements for Mordell-Weil groups of non{-}CM elliptic curves and some higher dimensional abelian varieties defined over number fields, and also for odd dimensional K-groups of number fields.Comment: This is a revised version of the MPI preprint no. 14 from March 200

On a Hasse principle for Mordell-Weil groups

In this paper we establish a Hasse principle concerning the linear dependence over $\Z$ of nontorsion points in the Mordell-Weil group of an abelian variety over a number field.Comment: First draft written on October 29, 2007. Submitted for publicatio