By Matthew Morrow (CNRS - Sorbonne Université)
Abstract: This expository survey will aim to provide an introduction to Scholze’s formalism of tilting, which serves as a sort of transfer principle through which p-adic problems in arithmetic geometry can be studied via characteristic p methods, without any requirement that p be large enough. Its simplest manifestation, namely the tilting correspondence for perfectoid fields, is a form of the classical field of norms construction of Fontaine and Wintenberger concerning infinitely ramified p-adic fields.
This yields valued fields in characteristic zero and p which “behave similarly”. But tilting goes far beyond the case of fields, eventually leading to the theories of perfectoid spaces and diamonds in which geometric objects of characteristic zero are embedded into a characteristic p world. It is a tantalizing question whether the resulting similar behaviour of objects in characteristic zero and p can be understood through model theory.