Elliptic curves and finitely generated Galois groups

Bo-Hae Im; Michael Larsen

REVIEW 2 major objections 4 minor 33 references

Elliptic curves and finitely generated Galois groups

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves that elliptic curves over fields with finitely generated Galois group have infinite rank.

desk verdict The genus-1 case of Junker–Koenigsmann is a major target and the paper's strategy is credible, but §5 has a real gap connecting the CRT residues to the Hales–Jewett point; the proof as written is not complete. read the letter →

arxiv 2510.00750 v2 pith:JRB7PJ54 submitted 2025-10-01 math.NT

Bo-Hae Im , Michael Larsen This is my paper · ORCID

classification math.NT MSC 11G0511G1014G0512E30

keywords ellipticcurvesinfiniterankGaloisgroupsfinitelygeneratedMordell-WeilHales-JewettChebotarevdensityabelianvarieties

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that if K is a field of characteristic zero whose absolute Galois group is finitely generated, then every elliptic curve over K has infinite Mordell–Weil rank. This answers an open conjecture from 2003 for elliptic curves, and it implies the genus-1 case of the Junker–Koenigsmann conjecture, which says that such fields are ample. The argument has two main parts: a Ramsey-theoretic construction, based on the Hales–Jewett theorem, produces many points on the elliptic curve over carefully chosen extensions; and a new 'Mordellic' theorem shows that the point group over the compositum of all bounded-degree extensions is virtually free, so any finitely generated subgroup must be free. A Chebotarev density step then shows these points generate a subgroup that grows without bound, forcing infinite rank.

What carries the argument

The main combinatorial engine is the Hales–Jewett theorem, which guarantees that any finite coloring of a high-dimensional cube contains a monochromatic combinatorial line. The authors use this to show that, after a finite Galois extension $L_0/K_0$, there is a finite collection of linear functions $t \mapsto a_i t + b_i$ such that for all but finitely many $t_0 \in L_0$, one of these functions gives the $u$-coordinate of a point on the affine model $y^2 = f(u)$ lying in the desired field $L$. The second key tool is a 'Mordellic' theorem (Theorem 3.1) asserting that for an abelian variety $A$ over a finitely generated field $K$, the group $A(K(d))$ of points over the compositum of all extensions of degree $\leq d$ is virtuall

What would settle it

A concrete way to refute the central claim would be to exhibit an elliptic curve over a characteristic-zero field with finitely generated absolute Galois group whose Mordell–Weil rank is finite; for instance, one might compute the rank of an elliptic curve over the fixed field of finitely many automorphisms of $\bar{Q}$ and find a finite value.

Watch

Extended reading notes

Core claim

For an elliptic curve $A_0$ over a finitely generated field $K_0$ of characteristic zero, and any finite set $\sigma_1,\ldots,\sigma_n$ of automorphisms of $\bar{K}_0/K_0$, the rank of $A_0$ over the fixed field $K_0^{\langle \sigma_1,\ldots,\sigma_n \rangle}$ is infinite. Equivalently, every elliptic curve over a field whose absolute Galois group is topologically finitely generated has infinite rank. The proof establishes two intermediate results: the points of an abelian variety over the extension generated by all of its torsion form a free abelian group modulo torsion, and the points over the compositum of all degree-$\leq d$ extensions form a virtually free group (finite torsion plus free abelian). These 'Mordellic' statements are then combined with the

Load-bearing premise

The proof requires that the Hales–Jewett construction, which is guaranteed to produce a point for some $t_0$ in the infinite field $L$, can be specialized to a $t_0$ in the finitely generated $\mathbb{Z}$-algebra $S_0$ while preserving the needed reduction properties; this step is asserted in §5 but not derived from the stated Theorem 4.2.

Editorial extensions

If this is right

Every elliptic curve over a field with finitely generated absolute Galois group has infinite rank.
The genus-1 case of the Junker–Koenigsmann conjecture holds: such fields are ample for pointed curves of genus 1.
The Mordellic theorem on bounded-degree extensions gives a new structural description of abelian variety point groups over large fields: virtually free, not just finitely generated.
The method combines Ramsey theory, Kummer theory, and Chebotarev density in a way that suggests strategies for the higher-dimensional case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the specialization step that takes t0 from L to S0 can be rigorously justified, the proof is complete; if a counterexample exists, it would likely involve a failure of the Hales–Jewett construction to interact well with reduction modulo primes.
A natural next step is to test whether the argument extends to abelian varieties of dimension > 1; if it does, the full 2003 conjecture and the full Junker–Koenigsmann conjecture would follow.
The virtually-free structure of A(K(d)) suggests that the rank of an abelian variety over large algebraic extensions can be studied via free group growth, potentially giving quantitative lower bounds on ranks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: this paper claims the elliptic curve case of the Junker–Koenigsmann conjecture, a genuinely important result. The strategy—Kummer theory over the torsion field, bounded-degree Mordell–Weil, Hales–Jewett, and a Chebotarev/finite-field counting argument—is coherent and builds sensibly on the authors' prior work.

What's new: previous results needed extra hypotheses (Heegner points, split quartic models, topologically cyclic fields). Theorem 1.1 removes them for all elliptic curves. The Kummer-theoretic part (Theorem 2.7) is interesting in its own right, and the proof of Theorem 3.1 is a nice combination of Silverman's lemma with Néron's theorem. The finite-field counting (Prop 5.2) is basically sound.

The main problem is in §5. After Proposition 5.2 fixes residues u_j mod m_j and CRT gives u0 ∈ S0, the proof simply says 'applying Theorem 4.2, we obtain t0 ∈ S0'—but Theorem 4.2, as stated, only produces t0 in L, and says nothing about reductions. The CRT element u0 is never used again. Without a congruence condition on t0, the reductions of the constructed point need not match the compatible tuple, so Proposition 5.2 cannot be invoked. This is a load-bearing gap, not a cosmetic one.

Secondary notes: Proposition 3.5 asserts Λ is free of finite rank by Theorem 2.7; that requires also noting W is finite-dimensional because A(K) is finitely generated. It's true, but the proof skips it. There are also notation slips (n vs k in the number of linear functions, a stray □ in Theorem 2.7).

The gap looks repairable: the Hales–Jewett construction works for every t0 in S0 after inverting finitely many denominators, the exceptional set is finite, and since there are finitely many colorings one can fix a linear function that works on an infinite CRT class. But that repair is not in the manuscript.

Who this is for: people working on anti-Mordellic questions and ample fields. It deserves a serious referee. The central claim is very likely true, but the submitted proof does not fully support it. I'd send it to review and ask for a revised §5, not desk-reject.

Referee Report

2 major / 4 minor

Summary. The paper claims Theorem 1.1: for an elliptic curve A0 over a finitely generated characteristic-zero field K0 and finitely many automorphisms σ_i of \bar K0, the rank of A0 over the invariant field \bar K0^{⟨σ_1,…,σ_n⟩} is infinite. This is shown to be equivalent to the statement that an elliptic curve over a field with finitely generated absolute Galois group has infinite rank, and to imply the genus-1 case of the Junker–Koenigsmann conjecture. The proof has three main components: (i) a Kummer-theoretic theorem (Thm 3.1) asserting that A(K(d)) is virtually free; (ii) a Hales–Jewett construction (Thm 4.2) producing, for all but finitely many t_0∈L, a u-coordinate from one of a finite list of linear forms that is the u-coordinate of a point on A(L); (iii) a Chebotarev/CRT argument in §5 intended to show that the traces of these points generate an infinite-rank subgroup of A(K). The final step is where the principal gap lies.

Significance. If correct, this is a major advance: it resolves Larsen's conjecture for elliptic curves and gives the genus-1 case of the Junker–Koenigsmann conjecture, removing the split-quartic hypothesis of the authors' earlier Hales–Jewett work. The high-level strategy is coherent and builds on deep known results (Bogomolov–Serre, Cadoret, Néron, Hales–Jewett, Chebotarev). The Kummer-theoretic part is largely self-contained modulo typos. However, the final Chebotarev step in §5 contains a load-bearing gap: the transition from the CRT element to the Hales–Jewett specialization is not justified. The gap appears repairable, but the result is not established by the current text.

major comments (2)

[§5, final proof of Theorem 1.1] After Proposition 5.2 gives a tuple and the CRT produces u_0∈S_0 with prescribed reductions, the text says: 'Applying Theorem 4.2, we obtain t_0∈S_0 such that ...' But Theorem 4.2 only asserts existence of t_0∈L; it gives no element of S_0 and no control of reductions modulo the m_j. The CRT element u_0 is never used. The later claim 'By the assumption concerning (u_1,…,u_k), these points in fact all lie in E(F_p)' requires that the reduction of the constructed u-coordinate be compatible with the tuple from Proposition 5.2. This is not established. The gap is plausibly repairable by choosing t_0∈S_0∩L in the congruence class defined by the CRT and using finiteness of the exceptional set in Theorem 4.2, but the proof as written is incomplete.
[§5, Proposition 5.2 application] The notation in the final paragraph is inconsistent: the finite linear system is indexed by k, while S_0 has degree n and there are n primes m_j; the tuple (u_1,…,u_k) from Proposition 5.2 is then used as if it supplied one residue per m_j. The linkage between the tuple, the primes, and the congruence class of t_0 is not stated. A correct proof would need to apply Proposition 5.2 to the n sequences Σ_j (the reductions modulo each m_j), obtain a tuple of length n, choose t_0 with the corresponding residues, and then verify that every conjugate of the selected point reduces to a compatible choice. Without this, the reduction argument cannot be checked.

minor comments (4)

[§2, Theorem 2.7] The statement says A(K), but the proof and the use in Proposition 3.5 require A(K_tor). As printed, the theorem is false: the torsion subgroup of A(K) is finite, not (Q/Z)^{2g}.
[§3, Proposition 3.5] The proof contains apparent typos: 'Let ℓ > n be a prime' should presumably read 'ℓ > d', and 'but not to K(A_tor)' should read 'but not to A(K_tor)'.
[§5] The sentence 'there exist infinitely many u_0∈S_0 whose (mod m_j) reduction is u_j for all j' is confusing and the subscript j is overloaded (primes vs Proposition 5.2 tuple coordinates). The element u_0 is never used afterwards.
[§4/§5] Theorem 4.2 is stated with coefficients in L, but the final proof asserts coefficients in L_0 and later assumes they lie in S_0. Either strengthen Theorem 4.2 or explain how the coefficients descend to L_0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorem is not presupposed by its inputs; the flagged §5 issue is a proof-gap/correctness concern, not a circular reduction.

full rationale

The derivation of Theorem 1.1 does not exhibit any of the circularity patterns. The Kummer-theoretic part (§2) is proved in the paper using Serre/Bogomolov's Theorem 2.1, the cohomological Propositions 2.3–2.5, and Néron's Mordell–Weil theorem; the reference to [L2] is introductory and the needed argument is supplied in the text. Theorem 3.1 uses [IL2, Proposition 6] only for a boundedness-of-torsion lemma and is otherwise proved from Propositions 3.2–3.5, Silverman's lemma, and Lemma 2.6. Theorem 4.2 is cited from [IL2]; this is a self-citation, but [IL2] is a published, parameter-free Hales–Jewett result that does not assume Theorem 1.1 and is used as an external lemma rather than as a renamed version of the target theorem. The Chebotarev/CRT argument in §5 is a new combination and does not fit data or rename a known result. The skeptical concern — 'Applying Theorem 4.2, we obtain t0 ∈ S0' when Theorem 4.2 as stated only guarantees t0 ∈ L — is a potential missing justification about integral specialization and reduction modulo m_j. That is a correctness or completeness issue, not circularity: even if the step fails as written, the proof would not be equivalent to its inputs by construction. No parameter is fitted, no prediction is defined in terms of the conclusion, and no load-bearing uniqueness claim is imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new postulated mathematical entities. All auxiliary objects (the constant c in Theorem 2.1, the integer m in Proposition 5.2, the ring S_0, etc.) are existential choices drawn from standard or cited theorems. The main external inputs are deep results from Galois representations and the authors' own prior publications; the central claim is not assumed by any of these inputs.

assumptions (8)

standard math Uniform bound on the index of the homothety subgroup in Z_ℓ^× for abelian varieties over finitely generated fields (Bogomolov-Serre, Theorem 2.1).
Imported from Serre's letters and [JJ]; used to prove Propositions 2.3 and 2.5.
standard math Independence of the K_{ℓ∞} fields for varying ℓ (Serre [S3], Cadoret [Ca]).
Used repeatedly to make cohomological arguments over products of Galois groups valid.
standard math Néron's generalization of the Mordell-Weil theorem to finitely generated fields of characteristic zero.
Used in Theorem 2.7 and Proposition 3.5 to conclude A(K_i) is finitely generated for finite extensions K_i.
domain assumption [IL2, Proposition 6]: a uniform bound on torsion of abelian varieties over degree ≤ d extensions of a finitely generated field.
Self-cited published result used in Proposition 3.4 to prove torsion finiteness in K(d).
standard math Hales-Jewett theorem on monochromatic combinatorial lines.
Used in §4 to construct u-coordinates producing points on A(L).
standard math Chebotarev density theorem for schemes over Z.
Used in §5 to ensure infinitely many primes p for which a suitable F_p-point of Spec S_0 exists.
standard math Riemann hypothesis for curves over finite fields (Weil bounds).
Used in Lemma 5.1 to count F_p-points on the curve X, giving a positive proportion of good u_0.
standard math Burnside's theorem on p-groups and Jordan's theorem on finite subgroups of GL_n(C).
Used in Proposition 3.3 to bound the size of subquotients of products of bounded-order groups.

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Pith. "Pith review of Elliptic curves and finitely generated Galois groups." pith.science (2026). https://pith.science/paper/JRB7PJ54

@misc{pith2026251000750,
  author       = {Pith},
  title        = {Pith review of: Elliptic curves and finitely generated Galois groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRB7PJ54}},
  note         = {Machine review of arXiv:2510.00750}
}

abstract

Let $K$ be an extension of $\mathbb{Q}$ and $A/K$ an elliptic curve. If $\mathrm{Gal}(\bar K/K)$ is finitely generated, then $A$ is of infinite rank over $K$. In particular, this implies the $g=1$ case of the Junker-Koenigsmann conjecture. This "anti-Mordellic'' result follows from a new "Mordellic'' theorem, which asserts that if $K_0$ is finitely generated over $\mathbb{Q}$, the points of an abelian variety $A_0/K_0$ over the compositum of all bounded-degree Galois extensions of $K_0$ form a virtually free abelian group. This, in turn, follows from a second Mordellic result, which asserts that the group of $A_0$ over the extension of $K_0$ defined by the torsion of $A_0(\bar K_0)$ is free modulo torsion.

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Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Geometric and differential Galois theories, 1--11, S\'emin

Bary-Soroker, Lior; Fehm, Arno: Open problems in the theory of ample fields. Geometric and differential Galois theories, 1--11, S\'emin. Congr., 27, Soc. Math. France, Paris, 2013

2013

[2] [2]

C.\ R.\ Acad.\ Sci.\ Paris S\'er

Bogomolov, Fedor Alekseivich: Sur l'alg\'ebricit\'e des repr\'esentations -adiques. C.\ R.\ Acad.\ Sci.\ Paris S\'er. A--B 290 (1980), no.\ 15, A701--A703

1980

[3] [3]

Canad.\ J.\ Math

Breuer, Florian; Im, Bo-Hae: Heegner points and the rank of elliptic curves over large extensions of global fields. Canad.\ J.\ Math. 60 (2008), no.\ 3, 481--490

2008

[4] [4]

Proc.\ London Math.\ Soc.\ (2) 13 (1913), 6--12

Burnside, William: On some properties of groups whose orders are powers of primes. Proc.\ London Math.\ Soc.\ (2) 13 (1913), 6--12

1913

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Cadoret, Anna: An open adelic image theorem for abelian schemes, IMRN 2015 (2015), 10208--10242

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Choi, Seokhyun; Im, Bo-Hae: Larsen's conjecture for elliptic curves over with analytic rank at most 1, arXiv:2502.18761

arXiv

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D.; du Sautoy, M

Dixon, J. D.; du Sautoy, M. P. F.; Mann, A.; Segal, D.: Analytic pro-p groups. Second edition. Cambridge Studies in Advanced Mathematics, 61. Cambridge University Press, Cambridge, 1999

1999

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Bull.\ Lond.\ Math.\ Soc

Dokchitser, Tim; Dokchitser, Vladimir: A note on Larsen's conjecture and ranks of elliptic curves. Bull.\ Lond.\ Math.\ Soc. 41 (2009), no.\ 6, 1002--1008

2009