First-order definability of Campana Points and Darmon Points in algebraic function fields in one variable over number fields

TL;DR

The paper develops a unified framework to define Campana and Darmon points in algebraic function fields in one variable over number fields by leveraging quadratic Pfister forms and the associated Witt-theoretic reciprocity. It reduces questions of definability to local-global data encoded by Pfister form residues across valuations, and establishes that Campana points admit a uniform ∀∃-definition while Darmon points admit a uniform ∀∃∀-definition (with the S=∅ case simplifying to ∀∃). Central technical ingredients include the introduction of valuation-encoded sets, the Reciprocity Theorem for Pfister forms, and a coordinate-polynomial transfer mechanism that preserves definability across finite extensions while controlling ramification. The results advance decidability and definability programs in global-function-field arithmetic by providing explicit, uniform first-order descriptions of these geometric point-sets and show how local valuation data aggregates into global definability statements. Together, these contributions pave the way for systematic first-order encodability of refined arithmetic subsets in function-field contexts, with potential impact on related counting and decision problems.

Abstract

We give first-order definitions of Campana and Darmon points in algebraic function fields in one variable over number fields. These sets are geometric generalizations of $n$-full integers (integers whose nonzero valuations are at least $n$) and perfect $n$th powers, respectively, to more general algebraic varieties. For this we exploit the theory of quadratic Pfister forms, which were used by Becher, Daans & Dittmann to extend to the case of algebraic function fields in one variable the methods used by Koenigsmann when proving that $\mathbb{Z}$ is universally defined in $\mathbb{Q}$. These methods had already been generalized to arbitrary global fields by Park (2013) and Eisenträger & Morrison (2018), and the author had already exploited these methods to find first-order definitions of Campana points (2024) and Darmon points (2024, with Handley) in the context of number fields. With the newly expanded version of these methods, we now transfer those results to the new context of algebraic function fields in one variable over number fields.

Theorems & Definitions (47)

• Theorem 1.1: \ref{['mainthmcampana']} and \ref{['mainthmdarmon']}
• Conjecture 1.2: Kollár
• Proposition 1.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 4.1
• Theorem 4.2: Witt Decomposition Theorem
• Definition 4.3
• Definition 4.4