On Larsen's conjecture on the ranks of Elliptic Curves

A. Hadavand

REVIEW 2 major objections 4 minor 1 cited by

On Larsen's conjecture on the ranks of Elliptic Curves

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves Larsen's conjecture for elliptic curves over Q when the generators of a finitely generated Galois group come from one of the paper's infinite families of Galois automorphisms.

desk verdict Neat fixed-point trick for Heegner points under involutions, but the independence theorem it leans on is mis-cited (CM vs arbitrary E), leaving the main theorem unsupported as written. read the letter →

arxiv 2411.14097 v1 pith:SC2VTBAM submitted 2024-11-21 math.NT

A. Hadavand This is my paper · ORCID

classification math.NT MSC 11G0511G1811R2911R37

keywords ellipticcurvesLarsen'sconjectureMordell-WeilrankHeegnerpointsHilbertclassfieldsGaloisfixedmodularparametrization

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

Larsen's conjecture predicts that for every elliptic curve $E$ over $\mathbb{Q}$ and every finitely generated subgroup $G$ of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the group of rational points on $E$ over the fixed field $\overline{\mathbb{Q}}^G$ has infinite rank. The paper proves the conjecture for the case in which each generator of $G$ belongs to one fixed infinite family $\Sigma_j$ of Galois automorphisms, families built by prescribing an involution on each Hilbert class field in an infinite family attached to imaginary quadratic fields of odd class number. The strategy produces an infinite set of independent Heegner points that are individually fixed by every generator of $G$, so their images on $E$ form an infinite independent set in $E(H_E^G)$. Because $H_E^G$ sits inside $\overline{\mathbb{Q}}^G$, the same conclusion transfers to $E(\overline{\mathbb{Q}}^G)$, giving Larsen's conjecture for these groups.

What carries the argument

The central mechanism is a family of Heegner points living in 'broad' Hilbert class fields $H_p$ rather than in a tower of 'deep' ring class fields. For each prime $p$ in the infinite set $A_N$, the imaginary quadratic field $k_p = \mathbb{Q}(\sqrt{-p})$ has odd class number $h_p$ and satisfies the Heegner hypothesis for the conductor $N$, so $\operatorname{Gal}(H_p/\mathbb{Q})$ contains exactly $h_p$ involutions. Fixing one index $j$ selects an involution $\psi_{jp}$ on every $H_p$, and the oddness of $h_p$ forces any such involution to fix at least one of the $h_p$ Heegner points $y_{jp}$ on $X_0(N)$ attached to $k_p$. The modular parametrization $\Phi_E : X_0(N) \to E$ sends each $y_{jp}$ to a Heegner point $P_p$ on $E$ defined over $H_p$, and an independence theorem for Heegner points attached to distinct imaginary quadratic fields ensures that, once the odd class numbers exceed a constant $C(E,\Phi_E)$, the points $P_p$ are non-torsion and independent. Since every generator $\sigma_i$ of $G$ restricts to $\psi_{jp}$ on $H_p$, all $y_{jp}$ are fixed by $G$, hence all $P_p$ lie in $E(H_E^G)$, forcing infinite rank.

What would settle it

Find two primes $p,q \in A_N$ for which the chosen involutions $\psi_{jp}$ and $\psi_{jq}$ cannot be realized simultaneously by any automorphism of the compositum $H_pH_q$; then $\Sigma_j$ is empty and Theorem 3.5 has no instances. A concrete route is to compute the Galois group of $H_pH_q/\mathbb{Q}$ and check whether the local prescriptions agree on the intersection of the two fields.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.5. For an elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$, if $G = \langle\sigma_1,\dots,\sigma_n\rangle$ is a finitely generated subgroup of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ whose generators all lie in one family $\Sigma_j$, then the rank of $E(H_E^G)$ is infinite; here $H_E$ is the compositum of the Hilbert class fields $H_p$ attached to the infinite family of imaginary quadratic fields $\mathbb{Q}(\sqrt{-p})$ with $p \in A_N$. Since $H_E^G$ is a subfield of $\overline{\mathbb{Q}}^G$, the same infinite-rank conclusion holds for $E(\overline{\mathbb{Q}}^G)$, which is exactly Larsen's conjecture for these groups. The paper also records a general version (Theorem 3.9) for any infinite family of imaginary quadratic fields satisfying the Heegner hypothesis whose odd class numbers exceed a constant depending on $E$ and its modular parametrization, and it exhibits a nested chain of $G$-fixed subfields over each of which the Mordell-Weil rank is infinite.

Load-bearing premise

The entire construction depends on the existence of a single automorphism $\sigma \in \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ whose restriction to every Hilbert class field $H_p$ is the chosen involution $\psi_{jp}$; the paper asserts that the family $\Sigma_j$ is infinite without proving that these local involutions are compatible, so if no such $\sigma$ exists the main theorem is vacuous.

Editorial extensions

If this is right

For any elliptic curve $E/\mathbb{Q}$ and any finitely generated $G$ whose generators lie in a single family $\Sigma_j$, the Mordell-Weil group $E(\overline{\mathbb{Q}}^G)$ has infinite rank, so Larsen's conjecture holds for these groups.
The rank is infinite already over the much smaller field $H_E^G$, the $G$-fixed subfield of the compositum of the Hilbert class fields.
For every infinite subfamily $A' \subseteq A_N$, the rank of $E\left(\left(\prod_{p\in A'} H_p\right)^G\right)$ is infinite, so deleting finitely many primes does not destroy the result.
Removing the first few primes from $A_N$ produces a nested chain of $G$-fixed subfields inside $H_E^G$, and each of these subfields still carries a Mordell-Weil group of infinite rank.

Reading between the lines

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

The pivotal unverified step is the lifting of the local involutions $\psi_{jp}$ to a single global automorphism of $\overline{\mathbb{Q}}$; if this compatibility can be proved, the families $\Sigma_j$ become concrete rather than conditional, and the main theorem applies to actual groups.
The fixed-index restriction on $j$ appears to be an artifact of the proof: the argument only needs the generators to preserve the chosen Heegner points $y_{jp}$, so generators drawn from different families might work whenever the corresponding global automorphisms exist.
The same broad-field construction should transfer to other modular settings: any quotient of a modular curve with an independence theorem for its Heegner points would yield infinite rank over the corresponding fixed fields, so the method is not inherently limited to elliptic curves.

Share X Bluesky LinkedIn Reddit HN

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

Quick take: the kernel is real. The observation that an odd class number forces any involution of the Hilbert class field to fix a Heegner point is new and neat, and the resulting theorem for finitely generated groups whose generators all act by the same prescribed involution on each H_p is not in the cited literature. That is honest progress for a narrow multi-generator family, not a breakthrough.

The load-bearing soft spot is Theorem 3.4. It is stated for an arbitrary elliptic curve over Q and cites Theorem 1.1 of [Sah13], but the title of [Sah13] says "on CM elliptic curves." If that theorem has a CM hypothesis, it does not cover non-CM E, and the proof of Theorem 3.5 collapses for those curves. The statement also claims independence in E(Q)/E_tors, while the Heegner points P_i are generally not in E(Q); the application silently switches to E(∏ H_{p_i})/tors. That may be a typo, but combined with the CM issue the central independence claim is unsupported. If [RS07] actually contains the needed result for all E, the paper could be repaired by a different citation; a referee would need to check that.

The other gaps are minor. The assertion that Σ_j is nonempty is unproved, but a one-paragraph linear-disjointness argument (each H_p ramifies only at p) supplies the global automorphism, so that worry is overblown. The proof of Corollary 2.7(1) is garbled, though the statement is a standard dihedral fact. Lemma 3.2 and the Dirichlet construction look correct.

Net: the idea is worth engaging, but the paper as written does not support the theorem for arbitrary elliptic curves. I would not desk-reject it; it deserves a referee who knows the Heegner-independence literature and can adjudicate the [Sah13] vs [RS07] question. But I would not cite it yet.

Referee Report

2 major / 4 minor

Summary. The paper claims to prove Larsen's conjecture on the infinite rank of E(Qbar^G) for finitely generated subgroups G of Gal(Qbar/Q) when each generator belongs to one of certain infinite families Sigma_j of Galois automorphisms. The strategy is to take infinitely many imaginary quadratic fields k_p with odd class number satisfying the Heegner hypothesis for the conductor N, attach to each its Hilbert class field H_p, and use Heegner points on X0(N) that are fixed by a chosen involution on each H_p. The authors define Sigma_j as the set of global automorphisms whose restriction to every H_p is the j-th involution of Gal(H_p/Q), assert that Sigma_j is an infinite family, and then use a cited independence theorem for Heegner points to conclude that the image of the fixed Heegner points under a modular parametrization gives an infinite independent set in E(H_E^G), hence infinite rank. The central claim is therefore a partial result toward Larsen's conjecture for a special class of finitely generated groups.

Significance. If the proof were correct, the paper would provide a new and interesting partial result toward Larsen's conjecture, going beyond the cyclic case previously treated by Im and others. The idea of using a 'broad' family of Hilbert class fields instead of a single deep ring class field is attractive and could be a useful technique. The paper ships no code and no machine-checked proofs, but the strategy is conceptually clear. However, the significance is conditional on resolving two load-bearing issues: the nonemptiness of the families Sigma_j is asserted without proof, and the independence theorem cited as Theorem 3.4 is misstated and is drawn from a source whose title indicates a CM hypothesis, raising serious doubt about its applicability to arbitrary elliptic curves. These gaps currently prevent the paper from establishing its main theorem.

major comments (2)

[Theorem 3.4] Theorem 3.4 is not correctly stated and its proof by citation is not sufficient. The points P_i = Phi_E(y_i) are defined over the Hilbert class fields H_{k_i}, not over Q (Theorem 2.5), so the phrase 'independent in E(Q)/E_tors' is meaningless unless all P_i happen to lie in E(Q), which is false in general. The application in the proof of Theorem 3.5 uses independence in E(prod H_{p_i})/tors, so the theorem statement should be corrected accordingly. More seriously, the cited result, Theorem 1.1 in [Sah13], is titled 'On the independence of Heegner points on CM elliptic curves associated to distinct quadratic imaginary fields'; if that theorem indeed assumes E has complex multiplication, then it cannot be used to establish Theorem 3.4 for an arbitrary elliptic curve over Q, and the infinite-rank conclusion in Theorem 3.5 for non-CM E is unsupported. The authors must either quote the exact theorem and verify that its hypotheses hold for all E, or supply a correct reference or proof that covers arbitrary modular elliptic curves. This issue is load-bearing because the proof of Theorem 3.5 depends entirely on the independence of the family {P_p}.
[Section 3, definition of Sigma_j] The nonemptiness of Sigma_j is asserted without proof. After defining Sigma_j := {sigma in Gal(Qbar/Q) | sigma|_{H_p} = psi_{jp} for all p in A_N}, the manuscript states 'Sigma_j is an infinite family' but gives no argument that such a global automorphism exists. If no such sigma exists, then Theorem 3.5 and Corollary 3.7 are vacuous. The existence can be proved by noting that the Hilbert class fields H_p are linearly disjoint over Q (their discriminants are supported on distinct primes), so any choice of automorphisms psi_{jp} on each H_p is compatible on finite composita, and then a global automorphism exists by compactness of Gal(Qbar/Q). This argument should be included. Since the hypothesis of Theorem 3.5 requires sigma_i in Sigma_j, this is a load-bearing missing step.

minor comments (4)

[Corollary 2.7(1)] The proof of Corollary 2.7(1) is hard to follow as written, partly because the text 'sqrt(-d) or (tau sigma)|_k = 1' should read 'sqrt(-d), so (tau sigma)|_k = 1'. The computation is correct after this typo is fixed, but the presentation should be clarified.
[Theorem 3.5 proof] In the proof of Theorem 3.5, the phrase 'sigma_i(y_jp) = psi_{jp}(y_jp) = y_jp, or y_jp in H_p^G' should use 'hence' instead of 'or' to avoid confusion.
[Corollary 3.7] The notation 'H_E^G < Qbar^G' is inappropriate because the fixed fields are fields, not groups; use 'subset' or 'subfield' notation.
[Theorem 3.9] Theorem 3.9 is stated too vaguely: the 'infinite families Sigma' are not defined, and the proof merely says 'one can do the same as in the Proof of Theorem 3.5'. This theorem should either be made precise or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the proof uses external independence theorems and an explicitly defined family of Galois automorphisms; no fitted input is renamed as a prediction.

full rationale

The derivation chain is self-contained in the sense that every step is either an external theorem or an explicit construction. Theorem 3.5 assumes σ_i ∈ Σ_j, where Σ_j is defined by the restrictions σ|H_p = ψ_jp; Proposition 3.3 then shows each ψ_jp fixes a Heegner point y_jp because the class number is odd. The conclusion σ_i(y_jp)=y_jp is literally the definition of Σ_j, not a fitted or predicted quantity, and the rank statement follows from the independence of the infinite family {P_p} cited from [Şah13] and [RS07]. The paper contains no self-citations and no parameter fitting. Two concerns raised by the skeptical reader are real but are not circularity: the assertion that Σ_j is infinite is not proved (a nonemptiness/compatibility gap, and repairable by ramification arguments), and Theorem 3.4 may overstate the scope of [Şah13], whose title indicates a CM hypothesis; both are correctness risks, not cases of a conclusion reducing to its own input. The Note 3.6 limitation also explicitly acknowledges an unproved injectivity property rather than concealing it. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. The main load-bearing assumptions are the cited independence theorem and the unproved existence of the Galois automorphism families Sigma_j. The latter is an ad hoc assumption that is not justified.

assumptions (5)

standard math Modularity theorem: for every elliptic curve E over Q of conductor N there is a surjective morphism Phi_E: X0(N) -> E defined over Q.
Invoked in Theorem 2.5 and Theorem 3.5 to obtain Heegner points on E from Heegner points on X0(N). Cited to [DS05].
domain assumption The independence theorem of Sahinoglu (Theorem 3.4) holds for every elliptic curve over Q, including non-CM curves.
The paper states Theorem 3.4 for arbitrary elliptic curves but cites [Sah13], whose title mentions CM elliptic curves. The scope mismatch is not resolved in the paper.
ad hoc to paper For each j, the set Sigma_j is nonempty: there exists sigma in Gal(Qbar/Q) whose restriction to every H_p is the specified involution psi_jp.
Asserted without proof in Section 3. Requires compatibility of the involutions on the compositum of the H_p, which is not established.
standard math Dirichlet's theorem on primes in arithmetic progressions.
Used in Lemma 3.2 to construct infinitely many primes p with the required congruence conditions.
standard math Siegel's theorem: log h_p / log p -> 1/2 as p -> infinity.
Used to ensure all but finitely many p in A_N have class number exceeding the constant C from Theorem 3.4.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On Larsen's conjecture on the ranks of Elliptic Curves." pith.science (2026). https://pith.science/paper/SC2VTBAM

@misc{pith2026241114097,
  author       = {Pith},
  title        = {Pith review of: On Larsen's conjecture on the ranks of Elliptic Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SC2VTBAM}},
  note         = {Machine review of arXiv:2411.14097}
}

abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=\langle\sigma_1, \dots, \sigma_n\rangle$ be a finitely generated subgroup of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. Larsen's conjecture claims that the rank of the Mordell-Weil group $E(\overline{\mathbb{Q}}^G)$ is infinite where ${\overline{\mathbb Q}}^G$ is the $G$-fixed sub-field of $\overline{\mathbb Q}$. In this paper we prove the conjecture for the case in which $\sigma_i$ for each $i=1, \dots, n$ is an element of some infinite families of elements of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Elliptic curves and finitely generated Galois groups
math.NT 2025-10 conditional novelty 7.0 of 10

If the Galois group of a field is finitely generated, every elliptic curve over it has infinite Mordell–Weil rank.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages · cited by 1 Pith paper

[1]

Heegner points and the rank of elliptic curves over large extensions of global fields

Florian Breuer and Bo-Hae Im. Heegner points and the rank of elliptic curves over large extensions of global fields. Canadian Journal of Mathematics , 60(3):481--490, 2008

work page 2008
[2]

Primes of the Form x2+ ny2

David A Cox. Primes of the Form x2+ ny2 . John Wiley and Sons, 1989

work page 1989
[3]

Rational points on modular elliptic curves

Henri Darmon. Rational points on modular elliptic curves . Number 101. American Mathematical Soc., 2004

work page 2004
[4]

A note on larsen's conjecture and ranks of elliptic curves

Tim Dokchitser and Vladimir Dokchitser. A note on larsen's conjecture and ranks of elliptic curves. Bulletin of the London Mathematical Society , 41(6):1002--1008, 2009

work page 2009
[5]

A first course in modular forms , volume 228

Fred Diamond and Jerry Michael Shurman. A first course in modular forms , volume 228. Springer, 2005

2005
[6]

Abelian varieties over cyclic fields

Bo-Hae Im and Michael Larsen. Abelian varieties over cyclic fields. American journal of mathematics , 130(5):1195--1210, 2008

work page 2008
[7]

Abelian Varieties and Finitely Generated Galois Groups

Bo-Hae Im and Michael Larsen. Abelian varieties and finitely generated galois groups. arXiv preprint arXiv:1902.04011 , 2019

work page Pith review arXiv 1902
[8]

On products of quadratic twists and ranks of elliptic curves over large fields

Bo-Hae Im and \'A lvaro Lozano-Robledo. On products of quadratic twists and ranks of elliptic curves over large fields. Journal of the London Mathematical Society , 79(1):1--14, 2009

work page 2009

Show all 13 references

[9]

Heegner points and mordell-weil groups of elliptic curves over large fields

Bo-Hae Im. Heegner points and mordell-weil groups of elliptic curves over large fields. Transactions of the American Mathematical Society , 359(12):6143--6154, 2007

2007
[10]

Rank of elliptic curves over almost separably closed fields

Michael Larsen. Rank of elliptic curves over almost separably closed fields. Bulletin of the London Mathematical Society , 35(6):817--820, 2003

2003
[11]

On the product of twists of rank two and a conjecture of larsen

\'A lvaro Lozano-Robledo. On the product of twists of rank two and a conjecture of larsen. The Ramanujan Journal , 19:53--61, 2009

2009
[12]

On the independence of heegner points associated to distinct quadratic imaginary fields

Michael Rosen and Joseph H Silverman. On the independence of heegner points associated to distinct quadratic imaginary fields. Journal of Number Theory , 127(1):10--36, 2007

2007
[13]

On the independence of heegner points on cm elliptic curves associated to distinct quadratic imaginary fields

Hatice S ahino g lu. On the independence of heegner points on cm elliptic curves associated to distinct quadratic imaginary fields. Proceedings of the American Mathematical Society , 141(3):813--826, 2013

2013

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Heegner points and the rank of elliptic curves over large extensions of global fields

Florian Breuer and Bo-Hae Im. Heegner points and the rank of elliptic curves over large extensions of global fields. Canadian Journal of Mathematics , 60(3):481--490, 2008

work page 2008

[2] [2]

Primes of the Form x2+ ny2

David A Cox. Primes of the Form x2+ ny2 . John Wiley and Sons, 1989

work page 1989

[3] [3]

Rational points on modular elliptic curves

Henri Darmon. Rational points on modular elliptic curves . Number 101. American Mathematical Soc., 2004

work page 2004

[4] [4]

A note on larsen's conjecture and ranks of elliptic curves

Tim Dokchitser and Vladimir Dokchitser. A note on larsen's conjecture and ranks of elliptic curves. Bulletin of the London Mathematical Society , 41(6):1002--1008, 2009

work page 2009

[5] [5]

A first course in modular forms , volume 228

Fred Diamond and Jerry Michael Shurman. A first course in modular forms , volume 228. Springer, 2005

2005

[6] [6]

Abelian varieties over cyclic fields

Bo-Hae Im and Michael Larsen. Abelian varieties over cyclic fields. American journal of mathematics , 130(5):1195--1210, 2008

work page 2008

[7] [7]

Abelian Varieties and Finitely Generated Galois Groups

Bo-Hae Im and Michael Larsen. Abelian varieties and finitely generated galois groups. arXiv preprint arXiv:1902.04011 , 2019

work page Pith review arXiv 1902

[8] [8]

On products of quadratic twists and ranks of elliptic curves over large fields

Bo-Hae Im and \'A lvaro Lozano-Robledo. On products of quadratic twists and ranks of elliptic curves over large fields. Journal of the London Mathematical Society , 79(1):1--14, 2009

work page 2009