turkmath.org

As a part of global class field theory, we construct a reciprocity map that describes the unramified (resp. tame) étale fundamental group as a pro-completion of a suitable idele class group (resp. tame idele class group) for smooth curves over finite fields. These results were extended to higher-dimensional smooth varieties over finite fields by Kato-Saito (unramified case, in 1986) and Schmidt-Spiess (tame case, in 2000). We begin the talk by recalling these results.

The main focus of the talk is to work with smooth varieties over local fields. The class field theory over local fields is not as nice as that over finite fields. We discuss results in the unramified class field theory over local fields achieved in the period 1981–2015 by various mathematicians (Bloch, Saito, Jennsen, Forre, etc.). We then move to the main topic of the talk which is the tame class field theory over local fields and prove that the results in the tame case are similar to that in the case of unramified class field theory.

This talk will be based on a joint work with A. Krishna and J. Rathore.