Seminar des SFB/TRR 326 GAUSOn a Characterisation of Perfectoid Fields by Iwasawa Theory

With a p-adic representation of the Galois group of a p-adic field are associated the Bloch-Kato groups defined via p-adic Hodge theory. Iwasawa theory motivates the study of these Bloch-Kato groups over infinite algebraic extensions of the field of p-adic numbers.

Over perfectoid fields, several results (by Coates–Greenberg, Perrin–Riou, Berger, P. . . . ) state that the Bloch–Kato groups admit a simple description.

In this talk, we will present a reciprocal statement: the structure of the Bloch–Kato groups associated with certain crystalline representations characterises the algebraic extensions of the field of p-adic numbers whose completion are perfectoid fields. In particular, we will recover, via a different method, results by Coates and Greenberg for abelian varieties, and by Bondarko for p-divisible groups.