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Rank stability makes rings of integers diophantine

Bjorn Poonen

Abstract

The recent negative answer to Hilbert's tenth problem over rings of integers relies on a theorem that for every extension of number fields $L/K$, if there is an abelian variety $A$ over $K$ such that $0 < \operatorname{rank} A(K) = \operatorname{rank} A(L)$, then $\mathcal{O}_K$ is $\mathcal{O}_L$-diophantine. We present an alternative proof of this theorem and review how it is used.

Rank stability makes rings of integers diophantine

Abstract

The recent negative answer to Hilbert's tenth problem over rings of integers relies on a theorem that for every extension of number fields , if there is an abelian variety over such that , then is -diophantine. We present an alternative proof of this theorem and review how it is used.
Paper Structure (8 sections, 5 theorems, 4 equations)

This paper contains 8 sections, 5 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. History
  3. Outline
  4. Notation
  5. Diophantine sets
  6. Using congruences
  7. Weakly approximating Z
  8. Shlapentokh's reduction

Key Result

Lemma 2.1

The set $\mathcal{O}_K-\{0\}$ is $\mathcal{O}_K$-diophantine.

Theorems & Definitions (14)

  • Remark 1.1
  • Lemma 2.1: Denef-Lipshitz1978
  • proof
  • Example 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 4 more