Number Theory Seminar

Divisibility in K-groups and classical conjectures in Number Theory

Grzegorz Banaszak, Professor of Mathematics, Mathematics, Adam Mickiewicz University,

I will discuss divisibility and wild kernels in algebraic K-theory of number
fields $F$ and present basic results concerning the divisible elements in
K-groups. Without appealing to the Quillen-Lichtenbaum conjecture one can
prove that the group of divisible elements is isomorphic to the
corresponding group of \' etale divisible elements.
One can apply this result for the proof of the $lim^1$ analogue of the
Quillen-Lichtenbaum conjecture. One can also apply it 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 $\zeta_{F}(s).$ I will discuss the relation
of divisible elements with Kummer-Vandiver and Iwasawa conjectures.
I will also present recent results, joint with Cristian Popescu, concerning
Brummer-Stark conjecture and Galois equivariant Stickelberger splitting map
in Quillen localization sequence. The Stickelberger splitting map is a basic
tool to investigate the structure of the group of divisible element

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For more information, please contact Pei-Yu Tsai by email at [email protected] or visit http://math.caltech.edu/~numbertheory/.