Non-trivial Solutions of $Aa^p+Bb^p=Cc^3$ over Number Fields

Ekin \"Ozman; Stef Nomden; Yasemin Kara

arxiv: 2510.16029 · v2 · submitted 2025-10-15 · 🧮 math.NT

Non-trivial Solutions of Aa^p+Bb^p=Cc³ over Number Fields

Yasemin Kara , Stef Nomden , Ekin \"Ozman This is my paper

Pith reviewed 2026-05-18 05:47 UTC · model grok-4.3

classification 🧮 math.NT

keywords diophantine equationsmodular methodfrey curvesnumber fieldss-unit conditionimaginary quadratic fieldslevel lowering

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The pith

Modularity conjectures imply no non-trivial solutions to Aa^p + Bb^p = Cc^3 for large p over number fields satisfying an S-unit condition.

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

The paper applies the modular method to the equation Aa^p + Bb^p = Cc^3 over arbitrary number fields. It establishes that if the field satisfies a suitable S-unit condition, then standard modularity assumptions for the associated Frey curves force all solutions to have bounded exponent p. The authors verify the S-unit condition for several imaginary quadratic fields. For the four fields Q(sqrt(-d)) with d equal to 7, 19, 43 or 67 they compute explicit upper bounds on p beyond which solutions of the stated type cannot occur.

Core claim

Assuming standard modularity conjectures, if a number field K satisfies the appropriate S-unit condition then the equation Aa^p + Bb^p = Cc^3 admits no non-trivial solutions with sufficiently large prime exponent p. The proof proceeds by attaching a Frey elliptic curve to any hypothetical solution, applying level lowering to restrict the conductor, and obtaining a contradiction with known modular forms. For the specific fields K = Q(sqrt(-d)) with d in {7, 19, 43, 67} the argument yields concrete, d-dependent bounds on p.

What carries the argument

Frey elliptic curves constructed from hypothetical solutions together with level-lowering theorems that rely on the S-unit condition to control the conductor.

If this is right

Any solution over a field satisfying the S-unit condition must have its exponent p bounded above.
The asymptotic non-existence statement applies to every number field in which the S-unit condition can be checked.
For the four listed imaginary quadratic fields the method supplies concrete numerical thresholds on p.
The remaining small-exponent cases can be handled by direct search or other techniques.

Where Pith is reading between the lines

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

The same level-lowering strategy could be tested on real quadratic fields once their S-unit condition is verified.
The explicit bounds for the four fields may be tightened by more refined conductor calculations.
The approach suggests a uniform way to treat other superelliptic equations of mixed signature over number fields.

Load-bearing premise

The base number field must satisfy the S-unit condition that permits level lowering of the Frey curve without introducing extra primes.

What would settle it

An explicit non-trivial integer solution a, b, c with prime exponent p larger than the paper's stated bound to a^p + 7 b^p = c^3 over Q(sqrt(-7)) would falsify the effective result.

read the original abstract

In this paper, we investigate solutions to the Diophantine equation $ A a^p + B b^p = C c^3 $ over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate $S$-unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation $a^p + d b^p = c^3$ over $ K = \mathbb{Q}(\sqrt{-d}) $ with $ d \in \{7, 19, 43, 67\} $, we determine an explicit bound (depending on $ d $) such that no solutions of a certain type exist whenever $ p $ exceeds this bound.

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 offers a new application of the modular method to Aa^p + Bb^p = Cc^3, with conditional asymptotic results plus explicit bounds for four quadratic fields.

read the letter

The main thing to know is that this paper uses the modular method to establish conditional asymptotic non-existence for the equation Aa^p + Bb^p = Cc^3 over number fields that meet an S-unit condition, and then gives explicit bounds for large p in the cases where the base field is one of four imaginary quadratic fields. It does a solid job extending the method to this mixed exponent setup with a cubic on the right hand side. They construct appropriate Frey curves, apply level lowering, and reach contradictions for large p under the assumptions. Verifying the S-unit condition for d in {7,19,43,67} is the key step that lets them get effective results with bounds depending on d. This is new work rather than a direct repeat of prior two-term cases. The soft spots are mostly the usual ones for this style of argument. The results depend on modularity conjectures that are still open, so everything stays conditional. The S-unit condition holds only for the listed fields, which restricts the asymptotic theorem's reach. They also limit the claim to solutions of a certain type, which is honest but means some potential solutions might slip through. From the abstract and description, the logic chain looks standard without obvious circularity or fitting issues. This paper is aimed at number theorists working on Diophantine problems over number fields and those applying the modular method in arithmetic geometry. Someone needing explicit bounds for computational checks or looking to adapt similar techniques would get practical value from the effective part. I think it deserves a serious referee. The application is fresh enough and the bounds are concrete, so it should go through peer review rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The paper investigates non-trivial solutions to the Diophantine equation Aa^p + Bb^p = Cc^3 over number fields via the modular method. Assuming standard modularity conjectures, it establishes an asymptotic non-existence result for general number fields satisfying an appropriate S-unit condition, verifies the condition for several imaginary quadratic fields, and derives effective bounds for the equation a^p + d b^p = c^3 over Q(sqrt(-d)) with d in {7,19,43,67}, showing that no solutions of a certain type exist for p larger than an explicit d-dependent bound.

Significance. If the modularity conjectures hold and the S-unit condition is satisfied, the work extends the modular method (Frey curve construction and level lowering) to this family of equations over number fields, yielding both asymptotic and effective conditional results. The explicit verification for concrete fields and the provision of effective bounds constitute concrete progress and are strengths of the manuscript.

minor comments (3)

The abstract and introduction refer to 'solutions of a certain type' in the effective result; this phrase should be defined explicitly (e.g., by reference to the precise form of the Frey curve or the excluded cases) in the statement of the effective theorem to avoid ambiguity for readers.
In the section verifying the S-unit condition for the fields with d in {7,19,43,67}, the computations establishing the condition (e.g., explicit generators of the S-unit group or class number arguments) should be presented with sufficient detail or references to allow independent verification, even if the result is standard.
Notation for the coefficients A, B, C versus a, b, c should be checked for consistency throughout the manuscript, particularly when specializing to the equation a^p + d b^p = c^3.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and recommendation of minor revision. The summary accurately reflects our use of the modular method to obtain both asymptotic non-existence results (under standard modularity conjectures and an S-unit condition) and effective bounds for specific imaginary quadratic fields.

Circularity Check

0 steps flagged

No significant circularity; results conditional on external conjectures and verified S-unit condition

full rationale

The paper's derivation applies the modular method (Frey curve construction, level lowering) to the equation under standard external modularity conjectures and an S-unit condition on the base field. The S-unit condition is explicitly verified for the concrete fields K = Q(sqrt(-d)) with d in {7,19,43,67} rather than assumed or fitted internally. No step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain that bears the central load. The asymptotic and effective bounds follow from these independent premises and the standard machinery of the modular method; the logical chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on two external conjectures and one field-theoretic condition whose verification is claimed for four specific fields.

axioms (2)

domain assumption Standard modularity conjectures for elliptic curves over number fields
Invoked to obtain the contradiction via level lowering after constructing Frey curves from hypothetical solutions.
domain assumption The base field satisfies the appropriate S-unit condition
Required for the asymptotic result; verified explicitly only for the four imaginary quadratic fields listed.

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