First-order definability of Darmon points in number fields

TL;DR

This paper provides a first-order logical description of Darmon points over number fields, revealing their definability and structure within the framework of model theory, and improves the definability in special cases.

Contribution

It introduces a first-order definability framework for Darmon points over number fields, including uniform formulas and quantifier bounds, enhancing understanding of their logical complexity.

Findings

Darmon points are definable with a $orallorallorall$-first order formula.

In special cases, Darmon points are definable with a $orallorall$-formula.

Formulas are uniform across all possible sets S, with explicit quantifier and degree bounds.

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 Eisentr\"ager & 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.