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

Juan Pablo De Rasis

arxiv: 2510.17042 · v2 · submitted 2025-10-19 · 🧮 math.NT · math.AG· math.LO

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

Juan Pablo De Rasis This is my paper

Pith reviewed 2026-05-18 05:41 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.LO

keywords first-order definabilityCampana pointsDarmon pointsalgebraic function fieldsquadratic Pfister formsglobal fieldsnumber fieldsmodel theory

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The pith

Campana points and Darmon points admit first-order definitions in algebraic function fields over number fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes first-order definitions for Campana points and Darmon points within algebraic function fields in one variable over number fields. These points generalize concepts like n-full integers and perfect powers to varieties in a geometric way. The definitions are achieved by applying an expanded theory of quadratic Pfister forms that builds on earlier work for number fields and global fields. A reader might care because this bridges model theory with arithmetic geometry, making these points accessible to logical analysis in a broader class of fields.

Core claim

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 nth powers, respectively, to more general algebraic varieties. This is done by exploiting the theory of quadratic Pfister forms to transfer methods from the number field case.

What carries the argument

The theory of quadratic Pfister forms, used to extend first-order definability methods from number fields and global fields to algebraic function fields in one variable.

Load-bearing premise

The theory of quadratic Pfister forms as developed for global fields transfers without modification to algebraic function fields in one variable over number fields.

What would settle it

Observing that the first-order formula proposed for Campana points fails to select exactly the points with the required multiplicity conditions on a specific function field and variety would disprove the result.

read the original 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 Eisentr\"ager & 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper transfers first-order definability of Campana and Darmon points to function fields over number fields by expanding Pfister form methods from the author's prior number-field work.

read the letter

The main thing to know is that this paper carries the first-order definability results for Campana and Darmon points over from number fields to algebraic function fields in one variable over number fields. It does so by adapting the quadratic Pfister form techniques that Becher-Daans-Dittmann used for function fields and that the author had already applied to the number-field case in 2024 papers (one solo, one with Handley). The abstract makes clear that the author treats this as a direct transfer using a newly expanded version of the method rather than a routine citation exercise. That transfer is the actual new piece; nothing in the referenced prior literature had handled the mixed setting with a number-field constant field and a curve. The citation chain is clean and proportionate: it starts from Koenigsmann, moves through Park and Eisenträger-Morrison for global fields, and then to the function-field work, without overclaiming independence. The author is explicit about building on his own earlier results, which is fine here because those results are the base case being extended. The soft spot is exactly the one the stress-test flags. When the constant field is infinite, the valuations coming from the number field can interact with the geometric valuations on the function field in ways that might add extra definable sets or require adjustments to keep the Pfister-form encodings first-order and precise for n-fullness and perfect-power conditions. The abstract says the methods have been expanded to cover this, but without the explicit formulas or the verification that the encoding still isolates exactly the Campana and Darmon points, it is hard to judge how cleanly the transfer goes through. That is a normal point to check in review rather than a fatal gap. This is a paper for people already working on model-theoretic definability in global fields and on generalizations of integer definability to varieties. A reader who knows the Pfister-form literature will see the incremental progress immediately and can judge the adaptation for themselves. It is the kind of targeted extension that deserves a serious referee rather than a desk reject; the claim is narrow but the method is grounded in prior verified techniques, so the details are worth checking.

Referee Report

1 major / 2 minor

Summary. The paper claims to give first-order definitions of Campana points and Darmon points in algebraic function fields in one variable over number fields, by extending the quadratic Pfister forms techniques previously applied to number fields (author's 2024 works) and to function fields over finite or algebraically closed constants (Becher-Daans-Dittmann), with an expanded method version that transfers the encoding of n-fullness and perfect-power conditions.

Significance. If the transfer holds, the result would extend first-order definability results from global fields to their function-field analogs over number fields, strengthening the toolkit for studying definable sets in arithmetic geometry. The manuscript explicitly credits the foundational contributions of Koenigsmann, Park (2013), Eisenträger & Morrison (2018), Becher-Daans-Dittmann, and the author's prior papers, and the method expansion itself constitutes a reusable technical advance.

major comments (1)

The central claim rests on the assertion that the Pfister-form encoding of n-fullness and perfect powers remains first-order expressible after transfer to K(C) with K a number field (infinite constant field). The manuscript must supply an explicit verification that the new constant-field valuations do not introduce additional definable sets or obstruct the existing formulas; without this, the transfer step is load-bearing but currently under-supported.

minor comments (2)

Clarify in the introduction the precise differences between the 2024 number-field definitions and the new function-field versions, including any additional axioms or lemmas required for the constant-field case.
Add a short table or diagram summarizing the chain of citations (Koenigsmann → Park/Eisenträger-Morrison → Becher-Daans-Dittmann → author 2024 → current transfer) to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for stronger explicit support of the transfer argument. We address the major comment below.

read point-by-point responses

Referee: The central claim rests on the assertion that the Pfister-form encoding of n-fullness and perfect powers remains first-order expressible after transfer to K(C) with K a number field (infinite constant field). The manuscript must supply an explicit verification that the new constant-field valuations do not introduce additional definable sets or obstruct the existing formulas; without this, the transfer step is load-bearing but currently under-supported.

Authors: We agree that an explicit, self-contained verification of the effect of constant-field valuations strengthens the argument. The expanded method in Sections 2–3 adapts the Pfister-form encodings by treating places of the constant field K via the same local-global principles and anisotropy conditions used for the number-field case; the formulas for n-fullness and perfect powers are shown to be insensitive to these additional valuations because any potential interference would violate the first-order definability already established over K. To make this verification fully transparent and address the referee’s concern directly, we will insert a new subsection (approximately 3.4) that isolates the constant-field contribution, proves that no extraneous definable sets are introduced, and confirms the existing formulas remain first-order expressible. This constitutes a major revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; transfer to function fields is independent work

full rationale

The paper cites the author's 2024 results on number fields and prior works by Becher-Daans-Dittmann, Park, and Eisenträger-Morrison as the foundation for quadratic Pfister form techniques. It then claims to expand those methods and transfer the first-order definitions of Campana and Darmon points to algebraic function fields in one variable over number fields. This transfer constitutes new mathematical content rather than a reduction by construction, self-definition, or load-bearing self-citation chain. No equations, parameters, or definability statements in the provided text are shown to be equivalent to their inputs tautologically. The derivation remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard algebraic and logical background with no apparent free parameters or new postulated entities; the key reliance is on the established theory of quadratic Pfister forms.

axioms (1)

standard math Theory of quadratic Pfister forms as developed and used by Becher, Daans & Dittmann and subsequent extensions to global fields
Invoked explicitly in the abstract as the tool for extending definability methods to function fields.

pith-pipeline@v0.9.0 · 5728 in / 1248 out tokens · 42511 ms · 2026-05-18T05:41:07.085075+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We exploit the theory of quadratic Pfister forms... Reciprocity Theorem of the Witt group (Theorem 6.5)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: Campana points are uniformly ∀∃-definable and Darmon points are uniformly ∀∃∀-definable

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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