arxiv: 2605.13590 · v1 · submitted 2026-05-13 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

On Galois Embedding Problems Arising from 3-Torsion of Elliptic Curves

Jos\'e-A. G\'alvez , Joan-C. Lario

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Pith reviewed 2026-05-14 17:58 UTC · model grok-4.3

classification 🧮 math.NT

keywords Galois embedding problemselliptic curves3-torsionmod 3 Galois representationscyclotomic fields3-division fieldsinverse Galois problemnumber fields

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The pith

Solvability of Galois embedding problems from elliptic curve 3-torsion over Q equals the existence of infinitely many such curves with matching 3-division fields in the cyclotomic case.

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

The paper extends the correspondence between Galois embedding problems and the 3-torsion of elliptic curves defined over the rationals to every possible image of the associated mod 3 Galois representation. It establishes that, when working over a cyclotomic base field, these embedding problems admit solutions if and only if infinitely many elliptic curves exist whose 3-division fields realize those solutions. This link converts questions about the solvability of abstract Galois extensions into questions about the arithmetic geometry of elliptic curves over Q. The extension covers the complete list of possible images: the full general linear group GL2(F3) together with its subgroups SD16, D6, D4 and C2^2.

Core claim

We extend the correspondence between Galois embedding problems and 3-torsion of elliptic curves over Q to all possible images of the mod 3 Galois representations, namely GL2(F3), SD16, D6, D4 and C2^2. In the cyclotomic case we show that solvability of these embedding problems is equivalent to the existence of infinitely many elliptic curves whose 3-division fields provide the corresponding solutions.

What carries the argument

The 3-division field of an elliptic curve over Q, which serves as the splitting field that solves the associated embedding problem for each possible image of the mod 3 Galois representation.

If this is right

Solvability of each listed embedding problem becomes equivalent to the existence of an infinite family of elliptic curves with prescribed 3-torsion.
The full set of possible images of mod 3 representations can now be treated uniformly by the same correspondence.
Concrete solutions to the embedding problems can be produced by constructing or finding elliptic curves with the required 3-division fields.
Questions about the inverse Galois problem for these groups reduce in the cyclotomic setting to arithmetic questions about elliptic curves.

Where Pith is reading between the lines

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

The equivalence may supply a method to produce explicit infinite families of elliptic curves realizing each possible mod 3 image by first solving the corresponding embedding problem.
This approach could connect to the study of the distribution of 3-torsion fields and to questions about ranks or heights of elliptic curves with fixed Galois images.
Similar correspondences might exist for other small primes or for torsion structures on abelian varieties of higher dimension.

Load-bearing premise

The known correspondence between embedding problems and elliptic-curve 3-torsion extends without extra conditions to the full list of images GL2(F3), SD16, D6, D4 and C2^2.

What would settle it

An explicit embedding problem over a cyclotomic field that is solvable by Galois theory but for which only finitely many elliptic curves over Q have a 3-division field that realizes the solution, or conversely an infinite family of elliptic curves whose 3-division fields fail to solve a known solvable embedding problem.

read the original abstract

We study Galois embedding problems arising from the 3-torsion of elliptic curves defined over $\mathbb{Q}$, extending the correspondence to all possible images of mod 3 Galois representations; namely, $\operatorname{GL}_2(\mathbb{F}_3),SD_{16},D_6,D_4$ and $C_2^2$. In the cyclotomic case, we show that solvability of these embedding problems is equivalent to the existence of infinitely many elliptic curves whose 3-division fields provide the corresponding solutions.

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

Extends the known mod-3 embedding correspondence to all five images and states a clean cyclotomic equivalence, but the reducible cases D4 and C2^2 need explicit checks to confirm the bijection carries over without extra conditions.

read the letter

The main takeaway is that this paper takes the existing links between solvable Galois embedding problems and elliptic curves with given mod-3 image and extends them to the full list: GL2(F3), SD16, D6, D4, and C2^2. In the cyclotomic setting it claims solvability of the embedding problem is equivalent to the existence of infinitely many elliptic curves over Q whose 3-division fields realize the image. That equivalence is the part that could be most useful downstream for inverse Galois constructions. The uniform treatment across all five groups is the clear advance over prior work that stopped at the irreducible cases. The paper appears to organize the groups systematically and state the cyclotomic equivalence directly, which keeps the argument focused. The soft spot is in the reducible images. For D4 and C2^2 the 3-torsion module is reducible, so it is not automatic that every solution to the embedding problem arises as the full 3-division field of an elliptic curve (or the converse). The stress-test concern is real on the abstract alone: the correspondence may require an extra surjectivity or freeness condition on the Galois action that is not needed for the irreducible cases. If the paper verifies this explicitly with the earlier bijections, the claim stands; if it relies on an unstated extension, that section is the one a referee should press. This is specialized algebraic number theory aimed at people already working on mod-p images or small Galois realizations. A reader who knows the classification of mod-3 images and the prior embedding results will get the most value as a reference or extension point. It deserves a serious referee because the claims are concrete and the topic is active, even though the reducible cases will need close checking in review.

Referee Report

1 major / 1 minor

Summary. The manuscript examines Galois embedding problems associated with the 3-torsion of elliptic curves over Q. It extends the known correspondence between solvable embedding problems and mod-3 images to the full list of possible images GL_2(F_3), SD_16, D_6, D_4 and C_2^2. In the cyclotomic case the central result asserts that solvability of the embedding problem for each such image G is equivalent to the existence of infinitely many elliptic curves E/Q whose 3-division field realizes exactly the Galois action of G.

Significance. If the equivalence is established without hidden restrictions, the work supplies a Galois-theoretic criterion for the existence of elliptic curves with prescribed mod-3 images in the cyclotomic setting. This links inverse Galois problems directly to arithmetic statistics of elliptic curves and furnishes a uniform framework that covers both irreducible and reducible cases.

major comments (1)

[Abstract and the section treating the images C_2^2, D_4] The equivalence asserted for the reducible images C_2^2 and D_4 (abstract) requires that every solution of the embedding problem arises as the 3-division field of an elliptic curve. Because the 3-torsion module is reducible for these images, an explicit verification is needed that no extraneous fixed fields appear; this step is load-bearing for the stated equivalence and should be isolated as a lemma.

minor comments (1)

[Introduction] Define the notation SD_16, D_6 and D_4 explicitly on first use and state the precise semidirect-product structures employed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below and will revise accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Abstract and the section treating the images C_2^2, D_4] The equivalence asserted for the reducible images C_2^2 and D_4 (abstract) requires that every solution of the embedding problem arises as the 3-division field of an elliptic curve. Because the 3-torsion module is reducible for these images, an explicit verification is needed that no extraneous fixed fields appear; this step is load-bearing for the stated equivalence and should be isolated as a lemma.

Authors: We agree that the claimed equivalence for the reducible images C_2^2 and D_4 requires an explicit check that every solution of the embedding problem arises precisely as the 3-division field of an elliptic curve over Q, without extraneous fixed fields introduced by the reducibility of the 3-torsion module. In the revised manuscript we will isolate this verification as a dedicated lemma in the section treating these images, thereby making the argument fully transparent and addressing the load-bearing step for the equivalence. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence derived from standard Galois correspondence without self-referential reduction

full rationale

The paper extends the known bijection between solvable embedding problems for mod-3 Galois images and elliptic curves over Q with given 3-torsion fields. The cyclotomic-case equivalence is stated as a theorem resting on external Galois-theoretic constructions (the correspondence for GL2(F3) and its subgroups) rather than any redefinition, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or definition in the provided abstract or description reduces the claimed solvability equivalence to its own inputs by construction. The extension to the listed images (including reducible cases C2^2 and D4) is presented as a direct verification within standard theory, not an ansatz smuggled via prior work by the same authors. This is the normal non-circular outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all such items remain unidentified without the full text.

pith-pipeline@v0.9.0 · 5386 in / 1126 out tokens · 50538 ms · 2026-05-14T17:58:08.484639+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/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery; embed_injective unclear

?

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

Theorem 4.1 … Galois embedding problem … solvable iff K is the splitting field of the 3-division polynomial of an elliptic curve E/Q
IndisputableMonolith/Foundation/AlexanderDuality.lean SphereAdmitsCircleLinking D ↔ D = 3 unclear

?

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

isomorphism Φ : S4 → PGL2(F3) … subgroups eGi ≅ GL2(F3), SD16, D6, D4, C2²

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

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