Student Number Theory: The Fontaine-Wintenberger theorem

Abstract: Let K be a perfectoid field of mixed characteristic (0, p). Then, one can form the tilt K^♭ of K, which is a perfectoid field of characteristic p. The Fontaine-Wintenberger theorem says that the absolute Galois group of K and the absolute Galois group of K^♭ are isomorphic. In this talk, we will define the tilting functor and construct its inverse using the Witt vectors. Then, we will see a more conceptual proof using the almost purity theorem. This proof also motivated Scholze to define perfectoid algebras.