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Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

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Pith's one-line read For every quadratic extension $K/F$ of number fields, the paper constructs an abelian variety $A/F$ of positive rank with $\operatorname{rank} A(F)=\operatorname{rank} A(K)$, yielding a negative answer to Hilbert's tenth problem over the…

desk verdict Strong new rank-stability theorem, but Lemma 2.4 has a real gap where inertness in K is used instead of inertness in F(√(qℓ)); the proof as written is incomplete. read the letter →

arxiv 2501.18774 v1 pith:HU3BPKMU submitted 2025-01-30 math.NT math.LO

Levent Alpöge , Manjul Bhargava , Wei Ho , Ari Shnidman This is my paper · ORCID
classification math.NTmath.LO MSC 11U0511G1014G0511R04
keywords Hilbert'stenthproblemrankstabilityabelianvarietiesSelmergroupssilentprimesquadraticextensionsdiophantinemodelsFermatcurves
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 for every quadratic extension $K/F$ of number fields there exists an abelian variety $A/F$ with positive rank whose rank does not grow when the base field is extended from $F$ to $K$: $\operatorname{rank} A(F)=\operatorname{rank} A(K)>0$. This is exactly the hypothesis that earlier work had shown would imply a negative answer to Hilbert's tenth problem over the ring of integers of every number field, so the paper completes that reduction. The proof produces the abelian variety as the Jacobian of a hyperelliptic curve $C_n:y^2=x^\ell+n$, controls it through its $(1-\zeta)$-Selmer group, and uses solutions to a $\Sigma$-unit equation to put a rational point on the right quadratic twist. If the proof is correct, no algorithm can decide whether a multivariable polynomial equation over $\mathcal{O}_K$ has a solution, for any number field $K$.

What carries the argument

The machinery is the $(1-\zeta)$-Selmer group of the Jacobian $J_n$ of $C_n:y^2=x^\ell+n$, where $\ell$ is an odd prime and $F$ contains $\zeta_\ell$; here $\phi=1-\zeta$ is a self-isogeny of degree $\ell$. A prime $\mathfrak{p}$ is silent when the local cohomology group $T_{\mathfrak{p}}=H^1(F_{\mathfrak{p}},J_n[\phi])$ vanishes, and Lemma 2.2 shows this happens whenever $\mathfrak{p}$ is inert or ramified in $F(\sqrt{n})$. Silence is what makes the Selmer group unchanged when $n$ is multiplied by $t^2$ for a $\Sigma$-unit $t$, so the rank-zero property of $J_{q\ell r^2}$ survives for $J_{q\ell r^2t^2}$. The paper combines this with the quadratic-twist rank identity $\operatorname{rank} J_{r^2a^{\ell-1}b^2}(K)=\operatorname{rank} J_{q\ell r^2a^{\ell-1}b^2}(F)+\operatorname{rank} J_{r^2a^{\ell-1}b^2}(F)$, in which the first summand vanishes.

What would settle it

For a concrete instance, take $F=\mathbb{Q}(\zeta_3)$, $\ell=3$, and $K=F(\sqrt{2})$, choose $r$ as in Yu's theorem, and compute $T_{\mathfrak{p}}=H^1(F_{\mathfrak{p}}, J_{6r^2}[\phi])$ at a prime $\mathfrak{p}$ inert in $K$; if $T_{\mathfrak{p}}\neq 0$ for any such $\mathfrak{p}$, the silent-prime lemma does not apply and the proof of Lemma 2.4 fails.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for any quadratic extension $K/F$ of number fields, there exists an abelian variety $A/F$ such that $\operatorname{rank} A(F)=\operatorname{rank} A(K)>0$. By the rank formula for quadratic twists, it is enough to find a Jacobian $J_{r^2a^{\ell-1}b^2}$ of positive rank over $F$ whose $K$-quadratic twist $J_{q\ell r^2a^{\ell-1}b^2}$ has rank zero. The paper obtains the rank-zero side from the $(1-\zeta)$-Selmer group: Yu's theorem supplies $r$ with $\operatorname{Sel}_\phi(J_{q\ell r^2})=0$, and the silent-prime lemma lets the Selmer group remain zero after multiplying the parameter by $t^2$, where $t=a^{(\ell-1)/2}b$ comes from a solution to the $\Sigma$-unit equation $a+2rb=1$. A twisted Fermat-curve cover then produces an $F$-rational point on $J_{r^2a^{\ell-1}b^2}$, and Lemma 2.6 guarantees it is non-torsion, so the rank is positive. Since $A:=J_{r^2a^{\ell-1}b^2}$ has the same positive rank over $F$ and $K$, Corollary 1.2 follows via the prior diophantine-stability result [MRS24].

Load-bearing premise

The load-bearing premise is that the primes left silent in Lemma 2.4—those inert or ramified in $K=F(\sqrt{q})$—are precisely the primes that Lemma 2.2 proves silent for the twists $J_{q\ell r^2t^2}$, because Lemma 2.2 only guarantees silence for primes inert or ramified in $F(\sqrt{q\ell})$, and the paper does not impose an extra hypothesis, such as $\ell \equiv 1 \bmod 4$ with $\sqrt{\ell}\in F$, that would make these two quadratic extensions coincide.

Editorial extensions

If this is right

  • Hilbert's tenth problem has a negative answer over $\mathcal{O}_K$ for every number field $K$: no algorithm can decide whether a multivariable polynomial equation over $\mathcal{O}_K$ has a solution.
  • $\mathbb{Z}$ has a diophantine model over $\mathcal{O}_K$ for every number field $K$, meaning the integers are existentially definable in the ring of integers in the sense used for undecidability transfers.
  • The result is unconditional; it does not depend on the finiteness of Tate-Shafarevich groups or any other unproved conjecture.
  • The specific Jacobians $J_{r^2a^{\ell-1}b^2}$ exhibit diophantine stability over $K/F$: their rational points do not grow under base change, matching the Mazur-Rubin notion of diophantine stability.
  • Together with the independent proof of Koymans and Pagano, the paper closes the last case of Hilbert's tenth problem over rings of integers of number fields.

Reading between the lines

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

  • A natural extension is to cyclic extensions of prime degree: the only local input is that primes inert in $F(\sqrt{n})$ are silent, so replacing the quadratic character by a cyclic character and keeping the same Fermat-curve Selmer setup may produce rank stability for such extensions. The authors prove the quadratic case only.
  • The proof is non-effective because Mitsui's theorem is a qualitative infinitude statement; an effective number-field circle method would convert the construction into an explicit family of abelian varieties with controlled conductor, which the paper does not address.
  • For fields $F$ that already contain $\zeta_3$ and for which $3$ is unramified in $K/F$, choosing $\ell=3$ makes the constructed abelian variety an elliptic curve; this is a direct dimension count from the genus formula, though the paper states the general abelian-variety version.
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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

2 major / 5 minor

Summary. The paper proves Theorem 1.1: for any quadratic extension K/F of number fields, there exists an abelian variety A/F with rank A(F) = rank A(K) > 0. By a theorem of Shlapentokh (in the form given in [MRS24]), this implies Corollary 1.2: Hilbert's tenth problem has a negative answer over the ring of integers of every number field. The proof works with the Jacobians J_n of the curves C_n: y^2 = x^l + n over a field F containing the l-th roots of unity, and studies the (1-zeta)-Selmer groups. It first uses a theorem of Yu to choose n = q l r^2 with vanishing Selmer group, then exploits silent primes to show that multiplying n by t^2 for suitable Sigma-units t does not change the Selmer group, and finally uses solutions of a Sigma-unit equation a + 2rb = 1, supplied by Mitsui's and Kai's results, to produce a rational point on the relevant quadratic twist. The construction is assembled in Section 4.

Significance. If the proof is completed, this is a landmark result: it would resolve, unconditionally and uniformly, a major open problem of the last few decades. The strategy is attractive and conceptually clean, combining Selmer-group methods (Mazur-Rubin, Yu) with additive combinatorics in number fields (Mitsui, Kai). The paper is also honest in attributing the reduction from rank stability to Diophantine undecidability to Shlapentokh, and the main claim is independently supported by the recent Koymans-Pagano proof of the corollary. The construction is parameter-free in the sense that it relies on external theorems rather than fitted data, and the main theorem is exactly the hypothesis needed for Shlapentokh's theorem, so the logical structure is transparent.

major comments (2)
  1. [§2.3, Lemma 2.4] The proof of the equality of local conditions W_p is incomplete for primes in S_inert. For n = q l r^2 t^2, we have F(√n) = F(√(q l)), so Lemma 2.2 yields T_p = 0 only when p is inert or ramified in F(√(q l))/F. However, S_inert is defined as the set of primes inert in K = F(√q). If p is inert in K but l is not a square in F_p^×, then p is split in F(√(q l)), so T_p is nontrivial; since p is in Sigma, t may have nonzero valuation at p, and the subsequent 'good reduction' clause cannot be invoked. Hence the assertion that Sel_φ(J_{q l r^2 t^2}) = Sel_φ(J_{q l r^2}) is not established. This is the only step forcing rank J_{q l r^2 t^2}(F) = 0 in Section 4, so the proof of Theorem 1.1 depends on repairing it. A natural repair is to choose l ≡ 1 (mod 4) before the Weil-restriction step in Section 4, so that √l ∈ F(ζ_l) and therefore F(√(q l)) = K for the reduced quadratic extension; with this hypothesis, every p inert in K is silent for J_{q l r^2 t^2}. Please state this condition and amend Lemma 2.4 and the proof accordingly.
  2. [§2.4, Proposition 2.7] The sentence 'Since a ≠ 0, this is not a torsion point' is not by itself a valid reason: on a curve of positive genus, a point with nonzero x-coordinate can represent a torsion class (for example on an elliptic curve). The intended argument must use the equality J_{r^2 a^{l-1} b^2}(F)_tors = J_{r^2 a^{l-1} b^2}[φ](F) obtained from Lemma 2.6 for all but finitely many t_{a,b}: because a ≠ 0, P is not fixed by the automorphism (x,y) ↦ (ζx,y), so the divisor class of P − ∞ is not fixed by ζ and hence is not in J[φ](F). Please make this reasoning explicit, since the positivity of rank in the final construction depends on it.
minor comments (5)
  1. [§2.3, notation] The term 'S_inert-unit' is used in the proof of Proposition 2.5 but is not formally defined; please define it explicitly, for instance as an element whose prime support is contained in S_inert.
  2. [§2.3, Lemma 2.3] The parenthetical justification involving Yu's assumption Gal(f) ≃ S_n is terse and potentially confusing: for f(x) = x^l + n over F(ζ_l), the Galois group is cyclic, not S_n. Please state precisely which theorem of Yu is being invoked and what condition on q corresponds to it.
  3. [§3, Proposition 3.1] The proof of Proposition 3.1 is presented as a sketch with references to Mitsui and Kai; since this proposition is a key input, please indicate the precise theorem or section of Mitsui (or of Kai) that supplies the three-prime result in the stated ideal-class form.
  4. [After Section 1] The quotation of Simon and Garfunkel's 'The Sound of Silence' is extraneous to a mathematics research paper; I recommend removing it.
  5. [§2.4, Lemma 2.6] The uniform bound on torsion and the choice of the integer N would be clearer if the text specified explicitly that N depends only on l and F, not on n or t.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a parameter-free derivation from external theorems, with no fitted input renamed as a prediction and no load-bearing self-citation.

full rationale

The derivation chain is self-contained against external results. Lemma 2.3 is taken directly from Yu's theorem, an independent published result, and Lemma 2.2 is proved from local Tate duality and Euler characteristic formulas. Proposition 2.5 is proved from Mitsui's number-field Goldbach theorem or Kai's linear-patterns theorem, both external and parameter-free. Proposition 2.7 uses Lemma 2.6, proved inline, and an explicit Fermat-curve covering. The final rank equality rank A_q(K) = rank A(F) + rank A_q(F) is a standard decomposition of the quadratic twist, not an input to the construction. The only cited work overlapping with the authors is Shnidman-Weiss [SW23], mentioned only as historical context among many unconditional results, and no conclusion depends on it. The silent-prime step in Lemma 2.4 flagged by the reader is a possible mathematical gap, namely whether a prime p inert in K automatically gives T_p = 0 for J_{qℓr²t²}, which Lemma 2.2 requires for F(√(qℓ)); but that is a correctness risk, not circularity. The lemma's asserted equality is not true by construction, and no fitted value or target theorem is assumed. Hence the circularity score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper relies on several substantial external theorems; the only ad hoc assumption is the unstated silent-prime condition needed in Lemma 2.4. No empirical fitting or invented entities appear.

free parameters (3)
  • ell (odd prime)
    Chosen as an odd prime not dividing Disc(K); the proof needs an additional condition such as ell ≡ 1 mod 4 so that √ell lies in F. The choice is load-bearing but not fitted to data.
  • r (twist parameter)
    Selected via Yu's theorem so that Sel_phi(J_{qℓr²}) = 0; existential, not fitted to data.
  • a, b (Sigma-units)
    Produced from infinite solutions to the Sigma-unit equation a + 2rb = 1; existential and not fitted.
assumptions (6)
  • domain assumption Shlapentokh's theorem (MRS24) reduces H10 for all number fields to the existence, for every quadratic extension K/F, of an abelian variety A/F with rank A(F) = rank A(K) > 0.
    Invoked in Section 1 to derive Corollary 1.2 from Theorem 1.1; not proved in this paper.
  • domain assumption Yu's Theorem 4 (Lemma 2.3): there exists r in O_F with Sel_phi(J_{qℓr²}) = 0 and r prime to the ramified primes of K.
    Used in Sections 2.3 and 4 to get a rank-zero twist; depends on q not being a square.
  • ad hoc to paper For p inert or ramified in K = F(√q), Lemma 2.4 asserts T_p = 0 for J_{qℓr²}; this needs p inert or ramified in F(√(qℓ))/F, which is automatic only when √ℓ ∈ F_p for such p, e.g., when √ℓ ∈ F.
    This is an unstated condition in Lemma 2.4's proof; without it the silent-prime set may be smaller.
  • domain assumption Mitsui's number-field Goldbach theorem (and/or Kai's theorem) supplies infinitely many prime elements p1, p2, p3 with p1 + β p2 = p3 and prescribed congruences.
    Used in Section 3 to prove Proposition 2.5; not reproved, only sketched.
  • standard math Serre-Tate potential good reduction and the uniform torsion bound in Lemma 2.6.
    Used to show all but finitely many twists have torsion equal to the phi-torsion.
  • standard math Weil restriction permits reduction from K(ζ_ell)/F(ζ_ell) to K/F.
    Section 4; standard technique but unproved in the paper.

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Pith. "Pith review of Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field." pith.science (2026). https://pith.science/paper/HU3BPKMU

@misc{pith2026250118774,
  author       = {Pith},
  title        = {Pith review of: Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HU3BPKMU}},
  note         = {Machine review of arXiv:2501.18774}
}
abstract

We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$.

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