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REVIEW 4 major objections 5 minor 22 references

Hilbert's 10th Problem via Mordell curves

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Pith's one-line read For five out of six primes, Hilbert's 10th problem is unsolvable over the cubic field Q(ζ3, ∛p); infinitely many degree-12 fields inherit the failure.

desk verdict A clean method note on H10 via cubic twists and cube-sum theorems, but the headline theorem is already subsumed by KP24/ABHS25; worth refereeing once the external theorems and two rank computations are verified. read the letter →

arxiv 2412.04253 v2 pith:JXQ2QZBD submitted 2024-12-05 math.NT

Somnath Jha , Debanjana Kundu , Dipramit Majumdar This is my paper · ORCID
classification math.NT MSC 11G0511U0511D25
keywords Hilbert'stenthproblemringsofintegersMordellcurvescube-sumcubictwistsrankstabilizationundecidabilitynumberfields
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

This paper aims to establish that for five out of every six primes $p$, Hilbert's 10th problem has a negative answer over the ring of integers of $L_p = \mathbb{Q}(\zeta_3, \sqrt[3]{p})$: no algorithm can decide whether polynomial equations with coefficients in that ring have solutions. The proof links this undecidability question to the classical cube-sum problem, which asks which integers are sums of two rational cubes and is encoded by the Mordell-Weil ranks of curves $y^2=x^3-432D^2$. A positive answer matters because it supplies a positive-density family of number fields where the analogue of Hilbert's 10th problem fails, using a cubic-twist mechanism rather than the quadratic-twist mechanism used in many earlier approaches. The paper also establishes that an infinite set of square-free integers $D$, counted by $X^{1-\varepsilon}$, produces the same failure over the larger field $\mathbb{Q}(\zeta_3, \sqrt{D}, \sqrt[3]{p})$ for every prime $p\equiv2,5\pmod9$.

What carries the argument

The carrying object is the Mordell curve $E_a:y^2=x^3+a$, together with its cubic twists $E_{aD^2}$ and $E_{aD^4}$, which are isomorphic to $E_a$ over $\mathbb{Q}(\zeta_3,\sqrt[3]{D})$. For the cube-sum connection, the relevant family is $E_{-432D^2}$, since for cube-free $D>2$ the torsion of $E_{-432D^2}(\mathbb{Q})$ vanishes and $D$ is a sum of two rational cubes exactly when this curve has positive rank. The argument is carried by the identity $$\operatorname{rk} E(L)=\operatorname{rk} E(K)+\operatorname{rk} E_{$D^{2}$}(K)+\operatorname{rk} E_{$D^{4}$}(K),\quad K=\mathbb{Q}(\zeta_3),\ L=K(\sqrt[3]{D}),$$ together with $\operatorname{rk} E_a(K)=2\operatorname{rk}E_a(\mathbb{Q})$. When exactly one of the three curves has positive rank over $\mathbb{Q}$, these identities force $\operatorname{rk}E(L)=\operatorname{rk}E(K)>0$, and the rank-retention criterion transfers the known unsolvability of Hilbert's 10th problem over $\mathbb{Q}(\zeta_3)$ up to $L$.

What would settle it

Compute the ranks of $E_{-432(\ell p)^2}$, $E_{-432(\ell p^2)^2}$, and $E_{-432(p\ell^2)^2}$ for $\ell=7,13,31,\ldots$ and primes $p\equiv8\pmod9$; any positive rank among these for infinitely many $p$ would directly refute the 100%-density lemma. Similarly, an independent computation showing $E_{-432\cdot 9^2}(\mathbb{Q})$ has rank zero would refute the $p\equiv2,5\pmod9$ case.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is a transfer from rank statements in the cube-sum problem to undecidability statements. The authors prove that Hilbert's 10th problem is unsolvable over the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{p})$ for every prime $p\equiv2,4,5,7\pmod9$, and for 100% of primes $p\equiv8\pmod9$ with respect to natural density, hence for $5/6$ of all primes. The transfer runs through the cluster of curves $E_a:y^2=x^3+a$, $E_{aD^2}$, and $E_{aD^4}$: when exactly one of these has positive rank over $\mathbb{Q}$, the rank of $E_a$ over $\mathbb{Q}(\zeta_3)$ is positive and is unchanged after adjoining $\sqrt[3]{D}$; a known rank-retention criterion then makes the integers definable by polynomial equations inside the larger ring, carrying unsolvability upward. For the degree-12 result, the proof shows that for an infinite square-free set $S$ with $\#(S\cap[-X,X])\gg X^{1-\varepsilon}$, the quadratic twist $E_{-432D^3}$ has positive rank, so the rank of $E_{-432}$ is positive over $\mathbb{Q}(\sqrt{D})$ and zero over $\mathbb{Q}(\zeta_3,\sqrt[3]{p})$ for $p\equiv2,5\pmod9$, and the quadratic-extension criterion delivers unsolvability over the compositum.

Load-bearing premise

The load-bearing premise is that the cited theorem saying certain products of two primes are not sums of two rational cubes still applies after the roles of the two primes are swapped, and that the fixed curve $y^2=x^3-432\cdot 9^2$ really has rank 1 over $\mathbb{Q}$; if the cubic-residue condition does not survive the swap, the density claim drops from 5/6 to 2/3, and if the rank-one computation fails, the $p\equiv2,5$ case breaks.

Editorial extensions

If this is right

  • For a set of primes of natural density $5/6$, the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{p})$ has an undecidable Diophantine problem, meaning no algorithm can decide solvability of polynomial equations there.
  • For every $D$ in an infinite square-free set $S$ with $\#(S\cap[-X,X])\gg X^{1-\varepsilon}$ and every prime $p\equiv2,5\pmod9$, the degree-12 field $\mathbb{Q}(\zeta_3,\sqrt{D},\sqrt[3]{p})$ also has an undecidable Diophantine problem.
  • Any new theorem saying that a family of integers is not a sum of two rational cubes can be fed into the same machine to produce new fields where Hilbert's 10th problem is unsolvable.
  • The paper supplies explicit congruence classes where the desired rank configuration provably occurs, so the undecidability statements are not merely existential.

Reading between the lines

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

  • Editorial inference: the sieve argument in the density proof should work for any finite list of auxiliary primes with the same rank-zero property, so the 100% density for $p\equiv8\pmod9$ likely extends to other residue classes as soon as the corresponding two-prime cube-sum theorems are available.
  • Editorial inference: combining the root-number computation mentioned in Remark 3.5 with a 3-Selmer parity condition would upgrade the lower bound for $S$ from $X^{1-\varepsilon}$ to a positive natural density, making the degree-12 family quantitative rather than sparse.
  • Editorial inference: the computational data in Section 2 point toward the stronger statement that for every cube-free $D$ some Mordell curve has exactly one positive-rank cubic twist; if true, unsolvability would hold over $\mathbb{Q}(\zeta_3,\sqrt[3]{D})$ for every cube-free $D$, not just a density-$5/6$ set.
  • Editorial inference: the same rank-transfer should iterate through higher layers of Kummer towers, turning a single-field undecidability result into undecidability for infinite towers of number fields.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper proves that Hilbert's 10th problem is unsolvable for the ring of integers of Q(ζ3, ∛p) for 5/6 of all primes p (Theorem 3.3), and for a family of degree-12 fields Q(ζ3, √D, ∛p) with D ranging over an infinite set S and p ≡ 2,5 (mod 9) (Proposition 3.4). The method combines Shlapentokh's rank-stabilization criterion with known results on the cube-sum problem: the rank of the relevant Mordell curves E_a is controlled by whether certain products of primes are sums of two rational cubes. The main theorem is reduced to three residue-class cases; the p ≡ 8 (mod 9) case is handled by a density argument using cube-sum theorems for products of two primes.

Significance. The paper's internal argument from rank conditions to unsolvability is clean and correctly applies Shlapentokh's theorem. The approach via cubic twists of Mordell curves is distinct from the quadratic-twist methods in [KP24] and [ABHS25], and the paper provides explicit computational data and raises natural open questions. The main theorem, however, is superseded by the recent announced proof of Hilbert's 10th problem for all number fields by Koymans and Pagano; the value of this note therefore lies in the method and the explicit families rather than in the statement of Theorem 3.3. The dependence on several external cube-sum theorems (some coauthored by the present authors) makes precise quoting and verification of hypotheses essential.

major comments (4)
  1. [Lemma 3.1(a)] The assertion that the elliptic curve E_{−432·9^2} has rank 1 over Q is unsupported; since Lemma 2.1 only requires positive rank, the authors should either exhibit and prove the non-torsion of an explicit rational point (such as (36,108)) or give a citation. Without this, the p ≡ 2,5 (mod 9) case is not established.
  2. [Lemma 3.2] The application of Theorem 1.1(b)(ii) requires the roles of the two primes to be interchanged relative to the statement as written: in the theorem the prime congruent to 8 mod 9 appears as ℓ, whereas in Lemma 3.2 it is the large prime p. The paper should state this relabeling explicitly and reproduce the exact hypotheses of [MS23, Thm. 1.1(b)(ii)], because if the published theorem carries any additional condition, the density-100% conclusion for primes p ≡ 8 (mod 9) is not justified.
  3. [Lemma 3.2] The sentence 'The set S is infinite' is asserted without proof; the infinitude of S = {ℓ prime : ℓ ≡ 4,7 (mod 9), 3 ∉ F_ℓ³} follows from Chebotarev's theorem but should be stated, as the density argument also relies on the fact that the congruence conditions p ∈ F_ℓ³ for distinct ℓ_i are independent with density 3^{-k}.
  4. [Proposition 3.4] The proof assumes rk_Z E_{−432}(Q(ζ3)) = 0 without proof; the curve E_{−432} has the rational point (12,36), which may be torsion, and the rank-zero claim needs a proof or citation. Additionally, the displayed equality 'rk_Z E_{−432p²}(Q) = rk_Z E_{−432p⁴}(Q)' should read that both ranks are zero; as written it is a tautology and does not support the subsequent conclusion.
minor comments (5)
  1. [Abstract] There is a typo in the abstract: 'tha Hilbert's 10th Problem' should be 'that Hilbert's 10th problem'.
  2. [Lemma 3.2] The indexing of the density steps is off by one: after processing ℓ₁,...,ℓ_k, the covered proportion is 1 − 3^{−k}, not 1 − 3^{−(k+1)}; the limit conclusion is unaffected.
  3. [Section 2.1] Equation (2.1) would benefit from a brief explanation that the equality of ranks over L follows from the fact that the three curves E, E₁, E₂ become isomorphic over L; the formula is used repeatedly in the proofs.
  4. [Section 2.2] It would be useful to state that the SAGE/MAGMA rank computations are rigorous (e.g., via 2-descent) for the curves listed, so that the examples are verifiable and reproducible.
  5. [Proposition 3.4] In the final paragraph, the notation 'A = E(D)' is confusing; it should say 'A = E_{−432D³}' (the quadratic twist).

Circularity Check

0 steps flagged · score 0.0 of 10

The proof is self-contained modulo externally proved cube-sum theorems; no circular reduction found.

full rationale

The derivation chain is not circular. Theorem 3.3 is assembled from Shlapentokh's criterion (Theorem 1.2), Lemma 2.1, and three cube-sum input theorems: Sylvester's classical non-cube-sum statement (Theorem 1.1(a)), the Dasgupta–Voight cube-sum criterion (Theorem 1.1(b)(i)), the [MS23] two-prime non-cube-sum theorem (Theorem 1.1(b)(ii)), and the [JMS23] AP cube-sum theorem (Theorem 1.1(c)). The latter two are authored in part by the present paper's authors, but they are external theorems with stated hypotheses that do not include the Hilbert-tenth target; the present paper does not fit parameters, rename fitted values as predictions, or define its objects in terms of its conclusions. The relabelling in Lemma 3.2, where the [MS23] parameters are used with the 8-mod-9 prime in the role of ℓ and the 4/7-mod-9 prime in the role of p, is a direct application of the quoted theorem and not a definitional identity. The computational examples in Section 2 are explicitly motivational and are not used in the proof of Theorem 3.3. The only unproved load-bearing assertion is 'The elliptic curve E_{−432∗9^2} has rank 1 over Q' in Lemma 3.1(a); that is a finite rank computation left unshown, which is an omitted proof rather than a circular step. The mention of Koymans–Pagano is contextual and not load-bearing. Therefore no claim reduces to its own input by construction, and the circularity score is 0.

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

The central result rests almost entirely on external and partly self-authored theorems about the cube-sum problem; the internal contribution is the observation that zero ranks of the relevant twists give rank stabilization. No parameters are fitted in the paper.

assumptions (5)
  • domain assumption Shlapentokh's rank-stability criterion (Theorem 1.2) and its quadratic-extension corollary (Theorem 1.3) are valid.
    The H10 conclusion follows by transitivity of Diophantine definability; accepted external result invoked in Lemmas 2.1 and 3.4.
  • domain assumption Theorem 1.1(a): for p ≡ 2, 5 (mod 9), p, p^2, 9p, and 9p^2 are not cube-sums (Sylvester).
    Used in Lemma 3.1(a) and Proposition 3.4 to force zero ranks of the relevant twists.
  • domain assumption Theorem 1.1(b): for p ≡ 4, 7 (mod 9) with 3 not in F_p^3, p is a cube-sum; and for ℓ ≡ 8 (mod 9) with the stated nonresidue condition, ℓp, ℓp^2, and pℓ^2 are not cube-sums (DV18, MS23).
    The main engine for both the p ≡ 4, 7 case and the density-100% lemma for p ≡ 8; exact hypotheses are critical.
  • domain assumption Theorem 1.1(c): for a ≡ 8 (mod 9) and gcd(a, d) = 1, infinitely many primes p ≡ a (mod 9d) are cube-sums (JMS23).
    Provides the auxiliary prime ℓ in Lemma 3.1(b).
  • ad hoc to paper The Mordell curve E_{-432 * 9^2} has rank 1 over Q.
    Asserted in Lemma 3.1(a) without proof or citation; a single rank computation that is load-bearing for the p ≡ 2, 5 (mod 9) case.

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Pith. "Pith review of Hilbert's 10th Problem via Mordell curves." pith.science (2026). https://pith.science/paper/JXQ2QZBD

@misc{pith2026241204253,
  author       = {Pith},
  title        = {Pith review of: Hilbert's 10th Problem via Mordell curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JXQ2QZBD}},
  note         = {Machine review of arXiv:2412.04253}
}
abstract

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\mathbb{Q}(\zeta_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof.

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