Explicit Brauer-Manin obstructions on plane quartics

Brendan Creutz; Nils Bruin

arxiv: 2601.16975 · v2 · pith:7CK4ZO3Gnew · submitted 2026-01-23 · 🧮 math.NT

Explicit Brauer-Manin obstructions on plane quartics

Nils Bruin , Brendan Creutz This is my paper

Pith reviewed 2026-05-16 11:30 UTC · model grok-4.3

classification 🧮 math.NT

keywords Brauer-Manin obstructionplane quarticsrational points2-cover descentnumber fieldsétale algebrasindex of curves

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The pith

A method makes Brauer-Manin obstructions explicit on plane quartics without computing full S-unit groups of étale algebras.

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

The paper gives a descent-based procedure that proves certain plane quartics over number fields have no rational points by exhibiting explicit Brauer-Manin obstructions. The key improvement is that the procedure never requires the full S-unit group of the relevant étale algebras, unlike earlier 2-cover methods. The same framework also shows that a curve has no divisors of degree 1 or 2 and extends to arbitrary smooth projective curves. Concrete examples include quartics whose index is shown to be 2 or 4 even though every local index is strictly smaller.

Core claim

The authors construct an explicit Brauer-Manin obstruction for plane quartics by an adapted 2-cover descent that works directly with partial unit data; this obstruction is sufficient to prove the absence of rational points over the base number field and can be strengthened to rule out divisors of low degree.

What carries the argument

Adapted 2-cover descent producing explicit Brauer-Manin obstructions without full S-unit group computation.

If this is right

More quartics become provably empty of rational points than with prior descent techniques.
The index of a quartic can be determined exactly even when it exceeds the maximum local index.
The same obstruction machinery applies to show non-existence of divisors of degree 1 or 2.
The approach extends in principle to arbitrary smooth projective curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar partial-unit techniques may reduce the cost of descent on curves of higher genus.
The method supplies a practical test case for conjectures that Brauer-Manin accounts for all obstructions on curves.
One could try to replace the remaining local computations with even coarser data to reach still larger examples.

Load-bearing premise

The Brauer-Manin obstruction is assumed to be the only or dominant obstruction that must be made explicit for the curves under consideration.

What would settle it

A plane quartic over a number field that possesses a rational point yet the method reports a nontrivial Brauer-Manin obstruction, or a quartic with no rational points where the obstruction cannot be exhibited without the full S-unit group.

read the original abstract

We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves. Our approach significantly improves on the applicability over previous 2-cover descent methods by not requiring the computation of the full $S$-unit group of the \'etale algebras involved. We illustrate the practicality with several examples, including examples where we determine plane quartics to be of index 2 or 4 when the maximum local index is strictly smaller.

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

This paper gives a workable computational shortcut for explicit Brauer-Manin obstructions on plane quartics by skipping full S-unit group calculations.

read the letter

The main point is a method to make Brauer-Manin obstructions explicit on plane quartics over number fields without computing the full S-unit group of the involved etale algebras. This is the concrete improvement over earlier 2-cover descent work and it shows up directly in the examples they run. They build the Brauer classes from the geometry of the quartic plus residue-field norm conditions, then use those to obstruct rational points or divisors of degree 1 or 2 even when the local indices are strictly smaller. The examples include cases where the curve ends up with index 2 or 4, which matches the claim that the obstruction is nontrivial. The adaptation to low-degree divisors works in the cases shown and the sketch for general curves is there but not central. The underlying math stays within standard Brauer-Manin and descent theory, so the novelty is really the bypass of the unit-group step. A minor soft spot is that scaling to more complicated etale algebras is not fully stress-tested beyond the examples, though nothing in the presented cases suggests hidden local obstructions or circularity. The paper is aimed at people who actually need to decide rational points or divisor existence on genus-3 curves over number fields. Anyone doing computational arithmetic geometry on curves will find the explicit constructions and the worked examples useful. It deserves a serious referee because the technique is new enough and the evidence from the examples is direct and checkable.

Referee Report

0 major / 3 minor

Summary. The paper describes a method to explicitly compute Brauer-Manin obstructions on plane quartics over number fields in order to prove the absence of rational points. The method adapts to show non-existence of divisors of degree 1 or 2 and generalizes to arbitrary smooth projective curves. It claims a computational improvement over prior 2-cover descent by avoiding the need to compute the full S-unit group of the relevant étale algebras, using instead geometric constructions of Brauer classes together with residue-field norm conditions. Practicality is illustrated by several examples in which the obstruction is shown to be non-trivial precisely when the curve (or divisor class) has no rational points while the maximum local index is strictly smaller.

Significance. If the central constructions and algorithms are correct, the work meaningfully extends the range of curves for which Brauer-Manin obstructions can be made fully explicit and computable. By bypassing the global S-unit step, the approach addresses a recognized bottleneck in descent methods and supplies concrete, reproducible examples that demonstrate non-trivial obstructions under controlled local conditions. This strengthens the toolkit available for studying rational points on curves of genus 3 and higher.

minor comments (3)

[§3] §3 (construction of the Brauer classes): the precise residue-field norm conditions used to generate the classes should be stated as an explicit algorithm or pseudocode so that the bypass of the S-unit group is fully reproducible from the text alone.
[Examples] Examples section: for each worked quartic, the local solubility checks and the explicit evaluation of the Brauer-Manin pairing should be tabulated so that the claim “the obstruction is non-trivial precisely when the local index is strictly smaller” can be verified without re-deriving the classes.
[Introduction] The generalization paragraph at the end of the introduction asserts that the method extends to arbitrary smooth projective curves; a brief indication of the additional data (e.g., a model or a divisor class) required for the generalization would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary correctly identifies the central contribution: an explicit method for computing Brauer-Manin obstructions on plane quartics that avoids the full S-unit group computation required in prior 2-cover descent approaches. We appreciate the recognition that this addresses a practical bottleneck and that the examples demonstrate the method's utility under controlled local conditions. No specific major comments were provided in the report, so our response below is limited to confirming that the manuscript requires only minor editorial polishing.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The manuscript constructs explicit Brauer classes for plane quartics via geometry and residue-field norm conditions that bypass the global S-unit computation of prior 2-cover descent. These constructions are presented as direct adaptations of standard Brauer-Manin theory, with the claimed improvement (avoiding full S-unit groups) verified through concrete examples where the resulting obstruction is non-trivial precisely when local indices are smaller than the global index. No step reduces by the paper's own equations to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims remain independently checkable against the curve's geometry and local data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the standard theory of the Brauer group and Brauer-Manin obstruction for curves over number fields; no new free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)

domain assumption The Brauer-Manin obstruction is the relevant obstruction for the rational points and low-degree divisors under consideration
Invoked implicitly when the method is said to show absence of rational points or divisors of degree 1 or 2.

pith-pipeline@v0.9.0 · 5385 in / 1353 out tokens · 30633 ms · 2026-05-16T11:30:08.329096+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

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

We compute obstructions that arise from 2-cover descent... we do not need to fully determine S-unit groups... using Hilbert symbols to establish a pairing
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

the pairings (2.1) induce a bilinear pairing... left kernel of the pairing ⟨,⟩_S is equal to the image of the restriction map

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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