Marco Artusa (CIRM (Luminy) and I2M (Marseille))

Duality theorems are among the central results in arithmetic geometry. For $p$-adic fields, the earliest example is due to Tate, dealing with Galois cohomology of finite Galois modules. To extend this result to more general coefficients, one is forced to modify the original cohomology groups. This underlines some shortcomings of Galois cohomology, such as the lack of a natural topology on cohomology groups. In this talk, we build a new topological cohomology theory for p-adic fields, thanks to the Weil group and Condensed Mathematics. Moreover, we see how to use this cohomology theory to extend Tate’s result to more general topological coefficients. This new duality takes the form of a Pontryagin duality between locally compact abelian groups. As a particular case, one gets the reciprocity isomorphisms of local class field theory “à la Weil”, which identifies the units of a $p$-adic field and the abelianised Weil group. One could try to apply similar techniques to higher local fields. Inspired by Kato’s work, the hope is to obtain a condensed-Weil version of higher local class field theory, which would identify $d$-th Milnor $K$-theory of a higher local field with its abelianised Weil group.