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Strongly minimal group relics of algebraically closed valued fields

Alf Onshuus, Assaf Hasson, Santiago Pinzon

Abstract

We prove Zilber's trichotomy for reducts of ACVF expanding $(K,+)$ or $(K^*, \cdot)$.

Strongly minimal group relics of algebraically closed valued fields

Abstract

We prove Zilber's trichotomy for reducts of ACVF expanding or .
Paper Structure (17 sections, 35 theorems, 99 equations)

This paper contains 17 sections, 35 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Background and preliminaries
  4. Preliminaries on ACVF
  5. Plane curves in strongly minimal structures
  6. A first reduction
  7. Complete models and analytic functions
  8. The setting
  9. Groups locally equivalent to $(K,+)$ or to $(K^*,\cdot)$
  10. On one dimensional algebraic groups.
  11. Locally affine groups
  12. Basic intersection theory in ACVF
  13. First order slopes and a first field configuration
  14. Power Series
  15. The additive case
  16. ...and 2 more sections

Key Result

Theorem 1

Let ${\mathcal{K}}$ be an algebraically closed valued field, and $\mathcal{G}:=(G,\oplus, \dots)$ a non-locally modular strongly minimal ${\mathcal{K}}$-group-relic. Assume further that $G$ is locally equivalent to either $(K,+)$ or to $(K^*,\cdot)$. Then $\mathcal{G}$ interprets a field, $F$, ${\ma

Theorems & Definitions (103)

  • Definition 1.1
  • Theorem 1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof
  • proof
  • Remark 2.6
  • Lemma 2.7
  • ...and 93 more