Forking in valued fields and related structures

TL;DR

This thesis advances the understanding of forking in valued fields, especially pseudo-local fields, by providing geometric descriptions, classification results, and conditions for non-forking types, connecting model theory with valuation theory.

Contribution

It offers a geometric description of forking in regular ordered Abelian groups and establishes an Ax-Kochen-Ershov principle for forking in valued fields, extending prior results.

Findings

Forking equals dividing in pseudo-local fields of residue characteristic zero.

Provided conditions for a parameter set to be an extension base in Henselian valued fields.

Connected forking behavior in valued fields with results on invariant measures.

Abstract

This thesis is a contribution to the model theory of valued fields. We study forking in valued fields and some of their reducts. We focus particularly on pseudo-local fields, the ultraproducts of residue characteristic zero of the p-adic valued fields. First, we look at the value groups of the valued fields we are interested in, the regular ordered Abelian groups. We establish for these ordered groups a geometric description of forking, as well as a full classification of the global extensions of a given type which are non-forking or invariant. Then, we prove an Ax-Kochen-Ershov principle for forking and dividing in expansions of pure short exact sequences of Abelian structures, as studied by Aschenbrenner-Chernikov-Gehret-Ziegler in their article about distality. This setting applies in particular to the leading-term structure of (expansions of) valued fields. Lastly, we give various sufficient conditions for a parameter set in a Henselian valued field of residue characteristic zero to be an extension base. In particular, we show that forking equals dividing in pseudo-local fields of residue characteristic zero. Additionally, we discuss results by Ealy-Haskell-Simon on forking in separated extensions of Henselian valued fields of residue characteristic zero. We contribute to the question in the setting of Abhyankar extensions, where we show that, with some additional conditions, if a type in a pseudo-local field does not fork, then there exists some global invariant Keisler measure whose support contains that type. This behavior is well-known in pseudo-finite fields.