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First-order definability of Darmon points in number fields

Juan Pablo De Rasis, Hunter Handley

TL;DR

The paper advances the understanding of first-order definability for Darmon points over number fields by constructing a uniform, first-order description of these points on $\mathbb{P}^1_K$ and showing a path to reduce quantifier complexity in special cases. It develops a coherent framework linking Campana orbifolds, quaternion algebras, and Diophantine sets to define Darmon points via valuations outside a fixed set of places, and provides explicit quantifier counts and degree bounds for the defining polynomials. Key contributions include a universal $ orall\exists\forall$-definition for the Darmon-point filtrations $D_{K,S,n}$, a simpler $\forall\exists$-definition when $S=\Omega_K^{\infty}$, and a detailed analysis of formula complexity, with conditional results toward further quantifier reduction. The work has implications for Hilbert’s Tenth Problem over number fields by offering concrete, parameter-uniform Diophantine descriptions that connect arithmetic geometry with logical definability, and lays groundwork for possible further refinements under additional uniformity assumptions.

Abstract

For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case. Darmon points, which are a geometric generalization of perfect powers, constitute a non-linear set-theoretical filtration between $K$ and its ring of $S$-integers, the latter of which can be defined with universal formulas, as has been progressively proven by Koenigsmann, Park, and Eisenträger & Morrison. We also show that our formulas are uniform with respect to all possible $S$, with a parameter-free uniformity, and we compute the number of quantifiers and a bound for the degree of the defining polynomial.

First-order definability of Darmon points in number fields

TL;DR

The paper advances the understanding of first-order definability for Darmon points over number fields by constructing a uniform, first-order description of these points on and showing a path to reduce quantifier complexity in special cases. It develops a coherent framework linking Campana orbifolds, quaternion algebras, and Diophantine sets to define Darmon points via valuations outside a fixed set of places, and provides explicit quantifier counts and degree bounds for the defining polynomials. Key contributions include a universal -definition for the Darmon-point filtrations , a simpler -definition when , and a detailed analysis of formula complexity, with conditional results toward further quantifier reduction. The work has implications for Hilbert’s Tenth Problem over number fields by offering concrete, parameter-uniform Diophantine descriptions that connect arithmetic geometry with logical definability, and lays groundwork for possible further refinements under additional uniformity assumptions.

Abstract

For a given number field , we give a -first order description of affine Darmon points over , and show that this can be improved to a -definition in a remarkable particular case. Darmon points, which are a geometric generalization of perfect powers, constitute a non-linear set-theoretical filtration between and its ring of -integers, the latter of which can be defined with universal formulas, as has been progressively proven by Koenigsmann, Park, and Eisenträger & Morrison. We also show that our formulas are uniform with respect to all possible , with a parameter-free uniformity, and we compute the number of quantifiers and a bound for the degree of the defining polynomial.
Paper Structure (14 sections, 25 theorems, 27 equations)

This paper contains 14 sections, 25 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Darmon Points
  4. FIRST EXAMPLE: $n$th powers
  5. MAIN EXAMPLE: Darmon Points over number fields
  6. Connecting Quaternion Algebras with Diophantine Sets
  7. Quaternion algebras
  8. Quadratic Hilbert Symbol
  9. Existentially and universally definable sets
  10. A General Method and First-Order Definitions of Darmon Points
  11. A method for constructing diophantine sets
  12. First-Order Definitions of Darmon Points
  13. Formula Complexity Calculations
  14. Sufficient conditions for quantifier improvement

Key Result

Theorem 1.1

Let $K$ be a number field and $S$ be a finite set of places of $K$ containing the archimedean ones. If $n\in\mathbb{Z}_{\geq 1}$, the set is $\forall \exists \forall$-definable in $K$, uniformly with respect to all possible such $S$. Moreover, the formula involves $2$ initial universal quantifiers, then $171$ existential quantifiers, and another $426$ universal quantifiers. The defining polynomial

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Proposition 4.1
  • Theorem 4.2
  • Definition 4.3
  • ...and 18 more