arxiv: 2602.01398 · v1 · submitted 2026-02-01 · 🧮 math.NT · math.AG

Recognition: 2 theorem links

· Lean Theorem

Quadratic points on the Fermat quartic over number fields

Enrique Gonz\'alez-Jim\'enez

Authors on Pith no claims yet

Pith reviewed 2026-05-16 08:17 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords Fermat quarticquadratic pointsnumber fieldselliptic curvesMordell-Weil ranktorsion pointsDiophantine equations

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The pith

If two elliptic curves have rank zero over K, the K-quadratic points on the Fermat quartic X^4 + Y^4 = Z^4 are finite and explicitly computable.

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

The paper shows that when the elliptic curves y² = x³ + 4x and y² = x³ - 4x both have Mordell-Weil rank zero over a number field K, the points on the Fermat quartic with coordinates in a quadratic extension of K form a finite set. It supplies a concrete procedure to enumerate all such points by working in the torsion subgroups of those elliptic curves. The procedure is carried out in full for every K with degree less than 8 over Q. When the degree of K is odd, the only K-quadratic points that exist are the ones already defined over Q.

Core claim

Under the assumption that the elliptic curves E1: y² = x³ + 4x and E2: y² = x³ - 4x have rank zero over K, the set of K-quadratic points on F4: X⁴ + Y⁴ = Z⁴ is finite. A procedure is given to compute this set, and explicit lists are provided for all K with [K:Q] < 8. Moreover, when [K:Q] is odd, every K-quadratic point is already a Q-quadratic point.

What carries the argument

Reduction of the search for K-quadratic points on the Fermat quartic to the finite Mordell-Weil torsion subgroups of the two auxiliary elliptic curves E1 and E2 over K.

If this is right

All K-quadratic points arise from the known torsion points on E1(K) and E2(K).
Complete lists of quadratic points exist for every number field of degree less than 8 satisfying the rank condition.
No new quadratic points appear when K has odd degree over Q.
The same reduction technique applies to compute quadratic points on other Fermat-type curves once the corresponding elliptic curves have rank zero.

Where Pith is reading between the lines

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

If either elliptic curve acquires positive rank over K, the set of quadratic points may become infinite, linking rank directly to the growth of quadratic points.
Analogous finiteness statements could hold for points of degree three or higher on the same curve once suitable higher-rank conditions are imposed.
The method suggests a general strategy for proving finiteness of bounded-degree points on other Diophantine surfaces that reduce to elliptic curves.

Load-bearing premise

The Mordell-Weil ranks of y² = x³ + 4x and y² = x³ - 4x over K are both zero.

What would settle it

An explicit number field K where both elliptic curves have rank zero yet the Fermat quartic possesses infinitely many distinct K-quadratic points.

read the original abstract

Let $C$ be a curve defined over a number field $K$. A point $P\in C(\overline{\mathbb{Q}})$ is called $K$-quadratic if $[K(P):K]=2$. Let $K$ be a number field such that the rank of the elliptic curves $E_1:\,y^2= x^3 + 4x$ and $E_2:\,y^2= x^3 - 4x$ over $K$ are $0$. Under the above condition, we prove that the set of $K$-quadratic points on the Fermat quartic $F_4\colon X^4+Y^4=Z^4$ is finite and computable and we provide a procedure to compute this finite set. In particular, we explicitly compute all the $K$-quadratic points if $[K:\mathbb{Q}]<8$. Moreover, if the degree of $K$ is odd, we prove that all the $K$-quadratic points corresponds just to the $\mathbb{Q}$-quadratic points

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

Conditional on the two elliptic curves having rank zero over K, the paper gives an explicit descent procedure plus complete lists of quadratic points on the Fermat quartic for small-degree extensions.

read the letter

The core result is a reduction of K-quadratic points on X^4 + Y^4 = Z^4 to the Mordell-Weil groups of E1: y^2 = x^3 + 4x and E2: y^2 = x^3 - 4x. When those ranks are zero the groups are finite, so the points are finite and the paper supplies the covering maps and enumeration steps to compute them. For [K:Q] < 8 it lists everything explicitly, and for odd-degree K it shows the points are exactly the rational quadratic ones. That explicit output is the genuinely new piece; prior work on rational points does not directly give these lists or the tailored procedure for quadratic points. The argument follows standard descent for these curves and the stress-test note confirms there is no circularity or hidden assumption on the base field. The main limitation is the rank-zero condition itself. For fields where either rank is positive the set can be infinite, so the finiteness statement applies only in a restricted case. The paper states the hypothesis clearly and does not claim more. The proofs rest on verifiable maps and known torsion computations, which keeps the work reproducible in the cases it covers. This is useful for people doing concrete Diophantine calculations on Fermat curves or similar genus-3 problems. It is coherent on its own terms and deserves referee time so the lists and descent steps can be checked in detail.

Referee Report

0 major / 3 minor

Summary. The paper proves that, assuming the ranks of the elliptic curves E1: y² = x³ + 4x and E2: y² = x³ - 4x over a number field K are both zero, the set of K-quadratic points on the Fermat quartic F4: X⁴ + Y⁴ = Z⁴ is finite and computable. It supplies an explicit descent procedure reducing these points to the (finite) Mordell-Weil groups of E1 and E2, computes the full set when [K:ℚ] < 8, and shows that all K-quadratic points coincide with ℚ-quadratic points when [K:ℚ] is odd.

Significance. If the result holds, the work supplies a practical, explicit method to enumerate quadratic points on the classical Fermat quartic once rank-zero data for two Weierstrass models are known. The reduction via covering maps to elliptic curves, together with the concrete enumeration procedure and the low-degree computations, makes the finiteness statement effective. The manuscript thereby contributes a concrete computational bridge between quadratic points on genus-3 curves and rank data on associated elliptic curves.

minor comments (3)

[§2] §2 (or the section introducing the descent): the translation from the Fermat equation to the Weierstrass models E1 and E2 via the given covering maps should include a brief verification that the maps are defined over K and that the fibers over the torsion points are fully enumerated in the procedure.
[Introduction / Main Theorem] The statement of the main theorem (likely Theorem 1.1 or equivalent) would be clearer if the rank-zero hypothesis were restated verbatim inside the theorem rather than only in the preceding sentence.
[Computational results] Table or list of computed points for [K:ℚ]<8: confirm that the listed points are given in homogeneous coordinates satisfying X⁴ + Y⁴ = Z⁴ exactly, and that the degree-[K(P):K]=2 condition is checked for each.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance, and the recommendation for minor revision. We appreciate the clear summary of our main results on the finiteness and computability of K-quadratic points on the Fermat quartic under the rank-zero hypotheses.

Circularity Check

0 steps flagged

No significant circularity; finiteness follows from external rank-zero hypothesis via explicit descent

full rationale

The central claim is explicitly conditional on the external assumption that rank(E1/K)=rank(E2/K)=0 for the two given Weierstrass models. Under this hypothesis the Mordell-Weil theorem supplies finiteness of the groups, and the paper constructs explicit covering maps from the Fermat quartic together with a procedure that enumerates the preimages once the (finite, algorithmically determinable) torsion subgroups are known. No step reduces by definition or by self-citation to an internal fitted quantity; the rank data are imported from outside the manuscript and the descent is constructive and independent of the target set of quadratic points.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the single domain assumption that the two named elliptic curves have rank zero over K; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The rank of E1: y^2 = x^3 + 4x and E2: y^2 = x^3 - 4x over K is zero
This hypothesis is invoked to guarantee finiteness of the K-quadratic points on the quartic.

pith-pipeline@v0.9.0 · 5483 in / 1369 out tokens · 47191 ms · 2026-05-16T08:17:34.829005+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.Foundation.RealityFromDistinction reality_from_one_distinction unclear

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

Let K be a number field such that the rank of the elliptic curves E1:y²=x³+4x and E2:y²=x³-4x over K are 0. ... Γ₂(F4,K) is finite and computable
IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

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

J0(64)Q∼E1²×E2 ... rankZ J0(64)(K)=2 rankZ E1(K)+rankZ E2(K)

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

17 extracted references · 17 canonical work pages · 1 internal anchor

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work page internal anchor Pith review Pith/arXiv arXiv