108 research outputs found

### 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 $K$-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

### 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

### 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

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