Contributions to the model theory of henselian fields

Thesis

The thesis addresses certain problems in the model theory of henselian fields, with a special focus on decidability. Some new methods are introduced along the way, which are of independent interest. The following results are obtained:

• A transfer theorem for perfectoid fields, saying that a perfectoid field K is decidable relative to its tilt Kb, modulo a subtle (albeit natural) condition on K. As an application, we prove that the fields Qp(p1/p∞) and Qp(ζp∞) admit decidable maximal immediate extensions, thereby obtaining some of the first few decidabilty results for tame fields of mixed characteristic.

• A model-theoretic proof of the Fontaine-Wintenberger theorem, which states that the absolute Galois groups of K and Kb are canonically isomorphic. (joint with F. Jahnke).

• A general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics, conditional to a certain form of resolution of singularities. This extends well-known existential Ax-Kochen/Ershov principles in residue characteristic 0 and also unramified mixed characteristic. It also encompasses the (conditional) existential decidability result of Denef-Schoutens for Fp((t)), which we also strengthen by replacing the assumption of (global) resolution with local uniformization.

• An undecidability result for the asymptotic theory of {K : [K : Qp] < ∞} in the language of valued fields with a cross-section. The proof goes via reduction to positive characteristic, à la Krasner-Kazhdan-Deligne, ultimately adapting Pheidas’ proof of the undecidability of Fp((t)) with a cross-section. This answers a variant of a question of Derakhshan-Macintyre.

• An undecidability result for power series fields k((t)), equipped with a total residue map res : k((t)) ! k, which picks out the constant term of the Laurent series. Becker-Denef-Lipschitz showed that res : k((t)) ! k is definable in the language of rings with a parameter for t, when the base field k is finite. We show that (k((t)),res) is undecidable, whenever k is infinite.

Authors: Kartas, K

• DOI: 10.5287/ora-548bggpxg
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: henselian fields; model theory; valued fields; perfectoid fields; resolution of singularities; Fontaine-Wintenberger
• Subjects: Geometry, Algebraic; Model theory; Algebraic number theory