On elementary classifications of perfectoid fields
Ferréol Lavaud

Abstract:
This thesis investigates the model theory of perfectoid fields, and revolves around the general project of classifying perfectoid fields, i.e. identifying an elementary class whose members match as closely as possible properties of perfectoid fields. We will investigate the question working in the language of valued fields, and also in the language of rings. Perfectoid fields are a class of valued fields whose original raison d’être was to transfer across characteristics. They have however proven extremely useful since, and have a number of applications, the most important one being a crucial advance in the weight-monodromy conjecture, a major problem of arithmetic geometry. See [18] for a survey.
Perfectoid fields may be defined in a number of different ways, depending on the choice of underlying language, most notably as valued fields, or as topological rings. These definitions are materially equivalent, in the sense that they describe the same class of structures, but they differ slightly in the restricted point of view of a working logician, as they are given in two different languages. For the sake of completeness, and in order to underline the richness of perfectoid fields, we will provide both points of view here, but we may already notice that the topological point of view shall be of little use to us - other than moral - as the language of topology is pretty adverse to first-orderisation attempts.
Our goal here is to find first-order analogues to the definitional properties of perfectoid fields, and axiomatise a class of models satisfying the analogue properties. A clear criterion of success will be that : 1. every perfectoid field should be a model of our theory, 2. every model of our theory should be elementarily equivalent to a perfectoid field. The characterisation up to elementary equivalence, rather than isomorphism, is a forced consequence of fundamental limitative results of model theory, for e.g. Löwenheim-Skolem theorems. At the cost of a relative lack of specificity, i.e. the existence of non-standard models, we aim to give an elementary theory, whose models still capture the most important properties of perfectoids.
The general plan of this thesis is as follows. The first chapter is a gentle introduction to perfectoid fields, mostly motivated as a bridge between characteristics. Chapters 2 and 3 present more model theoretic aspects of valued fields. The following chapters, which make the core of the thesis introduce a class of fields almost capturing perfectoids in the language of valued fields. The last chapter, the only one with a claim to originality, resolves a modified version of this classification problem, in the language of rings.