Seminar on Arithmetic Algebraic Geometry: Algebraic K-theory, Borel regulator, and Dedekind zeta function (1)

2012-10-11 10:00 - 2012-10-11 12:00
Faculty of Science Building #3 Room 413
Kei Hagihara
The analytic class number formula reveals the mysterious connection between the zeta function, which is onstructed from purely local arithmetic information, and the global arithmetic invariants, such as the ideal class group and the unit group, via a complex analytic meromorphic continuation.

Today, this fascinating formula is extremely generalised, although conjecturally, to more general motivic L-functions under the name of the Beilinson conjecture and the Bloch-Kato conjecture, and still attracts
many mathematicians.

In this talk, we survey Borel's theorems concerning these conjectures.
More concretely we give a sketch of the proof of the following two facts:

- the orders at integral arguments of the Dedekind zeta function of an algebraic number field are intimately related with the ranks of the corresponding algebraic K-groups.

- the "transcendental parts" of the leading terms at these arguments of the function are described in terms of the Borel regulator, which measures the discrepancy between the rational structure of algebraic K-groups and that of Beilinson-Deligne cohomology.