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Strongly Minimal Relics of T-convex Fields

Benjamin Castle, Assaf Hasson

Abstract

Generalizing previous work on algebraically closed valued fields (ACVF) and o-minimal fields, we study strongly minimal relics of real closed valued fields (RCVF), and more generally T-convex expansions of o-minimal fields. Our main result (replicating the o-minimal setting) is that non-locally modular strongly minimal definable relics of T-convex fields must be two-dimensional. We also continue our work on reducing the trichotomy for general relics of a structure to just the relics of certain distinguished sorts. To this end, we prove that the trichotomy for definable RCVF-relics implies the trichotomy for interpretable RCVF-relics, and also that the trichotomy for relics of o-minimal fields implies the trichotomy for relics of any dense o-minimal structure. Finally, we introduce the class of differentiable Hausdorff geometric fields (containing o-minimal fields and various valued fields), and give a general treatment of the trichotomy for one-dimensional relics of such fields (namely, reducing the trichotomy for one-dimensional relics to an axiomatic condition on the field itself).

Strongly Minimal Relics of T-convex Fields

Abstract

Generalizing previous work on algebraically closed valued fields (ACVF) and o-minimal fields, we study strongly minimal relics of real closed valued fields (RCVF), and more generally T-convex expansions of o-minimal fields. Our main result (replicating the o-minimal setting) is that non-locally modular strongly minimal definable relics of T-convex fields must be two-dimensional. We also continue our work on reducing the trichotomy for general relics of a structure to just the relics of certain distinguished sorts. To this end, we prove that the trichotomy for definable RCVF-relics implies the trichotomy for interpretable RCVF-relics, and also that the trichotomy for relics of o-minimal fields implies the trichotomy for relics of any dense o-minimal structure. Finally, we introduce the class of differentiable Hausdorff geometric fields (containing o-minimal fields and various valued fields), and give a general treatment of the trichotomy for one-dimensional relics of such fields (namely, reducing the trichotomy for one-dimensional relics to an axiomatic condition on the field itself).

Paper Structure

This paper contains 22 sections, 50 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Strongly Minimal Relics of Imaginary Sorts
  3. Reducing the O-minimal Restricted Trichotomy to Fields
  4. Interpretable RCVF-relics
  5. On Elimination of Certain Imaginaries
  6. Eliminating Interpretable Relics
  7. T-Convex Fields as Hausdorff Geometric Structures with a Lore
  8. Purity of Ramification
  9. A Topological Marikova Lemma
  10. Locally Manifold-Loric Groups
  11. Higher Dimensional Relics of T-Convex Fields
  12. Differentiable Hausdorff Geometric Fields
  13. Finding Infinitely Many Slopes
  14. TIMI in T-convex Fields
  15. Slopes
  16. ...and 7 more sections

Key Result

Theorem 1

Let $T$ be an o-minimal theory of fields, and $T_{conv}$ its $T$-convex expansion. Let ${\mathcal{R}}_V\models T_{conv}$, and let ${\mathcal{M}}=(M,...)$ be a strongly minimal non-locally modular definable ${\mathcal{R}}_V$-relic. Then $\dim(M)=2$.

Theorems & Definitions (145)

  • Conjecture 1.1: Peterzil
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 135 more