UC Berkeley

This thesis is concerned with developing a model theoretic understanding of henselian valued fields. The model theory of henselian valued fields has been a major topic of study during the last century, it was initiated by Robinson’s model completeness results for algebraically closed valued fields in [Rob56]. Remarkable work has been achieved by Haskell, Hrushovski and Macpherson to understand the model theory of algebraically closed valued fields, more precisely in [HHM05] and [HHM06] they clarify completely the picture for elimination of imaginaries showing that it is sufficient to add the geometric sorts. They also develop a no- tion of stable domination and independence in algebraically closed valued fields, that rather than being understood as a new form of stability should be grasp as a technique to lift ideas from stability to the setting of valued fields. This approach is for example illustrated in [HRK19], where notions of stability (e.g. germs, genericity, domintion, etc) are being used to give a complete description of definable abelian groups in algebraically closed valued fields.The starting point of this thesis relies on the Ax-Kochen principle, which states that the first order theory of a henselian valued field of equicharacteristic zero or of mixed char- acteristic, unramified and with perfect residue field is completely determined by the first order theory of its residue field and its value group. A natural principle follows from this theorem: model theoretic questions about the value group itself can be understood by reduc- ing them to questions into the residue field, the value group and their interaction in the field. A fruitful application of this principle has been achieved to describe the class of definable sets, see for example: [Pas90], [Bas91], [Kuh94]. The next natural step for understanding the model theory of henselian valued fields was obtaining an elimination of imaginaries statement. The first part of this thesis studies elimination of imaginaries in the setting of henselian valued fields of equicharacteristic zero with residue field algebraically closed. The obtained results are sensitive to the complexity of the value group, which is an ordered abelian group. In the first chapter we study elimination of imaginaries in ordered abelian groups, while in the second chapter we analyze imaginaries in henselian valued fields of equicharacteristic zero. The second part of this thesis studies domination results in an Ax-Kochen style in the setting of henselian valued fields of equicharacteristic zero. We use a more abstract notion of domination present in [EHM19] that generalizes the definition of stable domination present in [HHM05].