Alexander Schmidt

Аннотация: In this series of lectures I will give an introduction to Wiesend's class field theory for higher dimensional arithmetic schemes. It is more elementary than the class field theory of Bloch-Kato-Saito and therefore better suited for applications in some situations.

The setting is as follows: Given a regular scheme X, flat and of finite type over \mathit{Spec}({\mathbb Z}), there exists a higher idele class group \mathcal{C}_X which is build out of data attached to all curves on X. There is a natural reciprocity homomorphism

\[\mathit{rec}:\  \mathcal{C}_X \longrightarrow \pi_1^{et}(X)^{ab}\]

such that for every finite \'{e}tale Galois covering Y\to X the induced homomorphism

\[\mathcal{C}_X/N_{Y|X} (\mathcal{C}_Y) \longrightarrow Gal(Y|X)^{ab}\]

is an isomorphism. The higher idele class group carries a natural topology and, as in classical (i.e. one-dimensional) class field theory, the norm groups are exactly the open subgroups. Similar results hold for smooth varieties over finite fields, however, in this case the theory describes the tamely ramified coverings only.