Exact classification of elliptic curves $y^{2}=x^{3}-pqx$ with rank $0$ and trivial $\Sha[2]$

Arkabrata Ghosh; Paul M. Voutier

REVIEW 1 major objections 7 minor 9 references

Exact classification of elliptic curves $y^{2}=x^{3}-pqx$ with rank $0$ and trivial $\Sha[2]$

T0 review · 1 major / 7 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For the elliptic curves y^2 = x^3 - pqx with distinct odd primes p and q, the paper proves that both the Mordell-Weil rank over Q and the 2-torsion of the Tate-Shafarevich group vanish exactly when one of four explicit congruence conditions

desk verdict A real, non-circular complete classification for y^2 = x^3 - pqx; the only load-bearing gap is the imported evenness of dim C_S, which is probably fixable but should be spelled out. read the letter →

arxiv 2607.14033 v1 pith:3SAJF2A4 submitted 2026-07-15 math.NT

Arkabrata Ghosh , Paul M. Voutier This is my paper

classification math.NT MSC 11G0511G0714H52

keywords ellipticcurves2-isogenydescentSelmergroupsTate-Shafarevichgrouprankzerocongruenceconditionsquarticlocalsolubilityhomogeneousspaces

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 studies the family E_{p,q}: y^2 = x^3 - pqx, where p and q are distinct odd primes, and asks when the rank over Q is 0 and the 2-torsion of the Tate-Shafarevich group is trivial. It answers the question completely in elementary terms: both are zero if and only if exactly one of four congruence conditions holds, each involving p and q modulo 4, 8, or 16 and the Legendre symbol (p/q). The proof runs through a 2-isogeny descent, controlling the Selmer groups attached to the isogeny and its dual, together with a cokernel C_S that measures the failure of a natural map between them. If the classification is correct, it gives the first complete rank-zero/trivial-Sha[2] characterization for a composite-conductor family of this type, going beyond the previously understood single-prime case.

What carries the argument

The central object is the descent via the degree-2 isogeny φ from E_{p,q} to its isogenous curve, with homogeneous spaces C_d of the form w^2 = d u^4 + ... whose local solubility over Q_p is classified by an imported quartic criterion (Lemma 3.1). The φ- and φ̂-Selmer groups are subsets of Q(S,2) = {±1, ±2, ±p, ±q, ±2p, ±2q, ±pq, ±2pq} consisting of classes whose homogeneous space has points in every local field. The novel twist is the cokernel C_S of the map φ_S from the 2-Selmer group to the φ̂-Selmer group; C_S is nontrivial in general, and its evenness, imported from a higher-descent alternating pairing, is what allows the converse direction of Theorem 1.2 to go through.

What would settle it

Check the predicted Selmer groups for (p, q) = (7, 5), which satisfies condition (ii): a 2-isogeny descent should give exactly S(φ)(E/Q) = {1, 35} and S(φ̂)(Ê/Q) = {1, -35}, with C_S = 0. Any additional locally soluble homogeneous space in Q(S,2) would falsify the theorem's p = 2 analysis, as would a rank or Sha[2] computation contradicting Theorem 1.2 for this pair.

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Extended reading notes

Core claim

Theorem 1.2 is the paper's central claim: for distinct odd primes p and q, the curve E_{p,q}: y^2 = x^3 - pqx has rank 0 over Q and dim_{F2} Sha(E_{p,q}/Q)[2] = 0 if and only if one of four conditions holds: (i) p,q ≡ 1 (mod 4), pq ≡ 13 (mod 16) and (p/q) = -1; (ii) p not ≡ q (mod 4), pq ≡ 3 (mod 8) and (p/q) = -1; (iii) p ≡ q ≡ 3 (mod 4), (p/q) = 1, q ≡ 3 (mod 8) and p - q ≡ 8 or 12 (mod 16); (iv) p ≡ q ≡ 3 (mod 4), (p/q) = -1, p ≡ 3 (mod 8) and q - p ≡ 8 or 12 (mod 16). The proof shows that these are exactly the cases where the φ- and φ̂-Selmer groups both have minimal size, {1, pq} and {1, -pq}, and where the cokernel C_S of the induced Selmer map is trivial; a corollary then gives rank 0

Load-bearing premise

Every mod-8 and mod-16 condition in the theorem is produced by the p = 2 branch of an imported local-solvability classification for quartics ax^4 + by^4 + cz^2 = 0 over Q_2, and the paper gives no proof of that classification; if that branch is wrong or misapplied, the necessary-and-sufficient conditions would shift.

Editorial extensions

If this is right

Rank 0 and trivial 2-primary Sha for E_{p,q} become decidable by simple modular arithmetic and a Legendre symbol calculation, with no elliptic-curve computation needed.
Whenever one of the four conditions holds, the φ- and φ̂-Selmer groups are minimal and, by Corollary 1.5, the rank over Q(i) is also 0.
The analysis of C_S explains why earlier single-prime results do not extend naively: the dimension of rank plus Sha[2] is not just the sum of the two Selmer dimensions minus 2, but also depends on this cokernel.
If the remaining eight size possibilities for the two Selmer groups are worked out, the same machinery would characterize all possible values of rank + dim_2 Sha[2] for this family, fully generalizing the one-prime classification.
For pairs satisfying condition (ii) with p ≡ 7 and q ≡ 5 mod 8, the theorem strengthens the previously known rank-zero result by showing the 2-primary Sha is actually trivial.

Reading between the lines

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

The four congruence conditions likely imply an explicit positive proportion of prime pairs for which E_{p,q} has rank 0 and trivial 2-primary Sha; the Legendre symbol and congruence statistics are roughly independent, so a density estimate should be derivable.
The same descent-plus-quartic-local-solvability recipe, with the parity of C_S handled, should yield complete classifications for every family y^2 = x^3 - Dx where D is a product of primes; the bottleneck is not local solubility but the global cokernel.
The Q(i) corollary suggests a concrete analytic check: for primes satisfying the conditions, the complex L-function should have order 0 at s = 1, which would be a BSD-consistent verification of the rank statement.
A sharper computational probe is to compute C_S for pairs with dim S(φ̂) = 3, such as p, q ≡ 1 mod 4 with (p/q) = 1; the paper's converse relies on the evenness of C_S in exactly that case, so any odd-dimension example would expose a missing hypothesis.

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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

Short version: this paper delivers what it claims — an exact congruence classification of when rank is 0 and Sha[2] is trivial for the two-parameter family y^2 = x^3 - pqx. I spot-checked the descent machinery and the homogeneous-space lemmas, and everything checks out. The novelty is genuine: Silverman's theorem is one-prime, and Ghosh's earlier result is the p=7, q=5 mod 8 special case. Conditions (i)–(iv) are new, explicitly computable, and the proof is genuinely derived rather than fitted. The C_S cokernel formula (Proposition 2.1) is a nice reusable tool, and the snake-lemma argument connecting C_S to C_X is clean. The strengthening to trivial X[2^∞] over Ghosh's earlier work is also real.

The soft spots are real but not fatal. The main one is Lemma 2.2: the evenness of dim C_S is imported from Fisher's Theorem 2.1 with a two-sentence identification of his S_1, S_1', S_2, S_2' with the groups here. That lemma is load-bearing in the converse of Theorem 1.2, specifically in excluding the dim S(phi-hat)=3 case. I could not find an actual error — the curves do have rational 2-torsion, so Fisher's setup is very likely applicable — but the text does not verify his hypotheses or pin down the identifications. A referee should ask for that to be written out.

Two smaller issues. The introduction says C_S is nontrivial for some curves, which motivates the whole cokernel formalism, but no example is given. That is worth a sentence or a footnote. And there is no shipped computational cross-check of the four congruence conditions, even though the authors do cite PARI/GP computations for rank. A short verification script or table would substantially raise confidence in the Section 4 case analysis.

Overall: the central argument holds up, the classification is new and clearly stated, and the paper deserves a serious referee. The issues are fixable in revision. If I worked on 2-descent or Selmer groups for these families, I would cite it and would bring it to a reading group.

Referee Report

1 major / 7 minor

Summary. The paper studies the elliptic curves E_{p,q}: y^2 = x^3 - pq x for distinct odd primes p,q. It gives four elementary congruence conditions, in terms of Legendre symbols and mod-8/mod-16 residues, that are equivalent to both the φ- and φ̂-Selmer groups being of minimal size (Theorem 1.1). It then proves that these same conditions are equivalent to rank E_{p,q}(Q) = 0 and X(E_{p,q}/Q)[2] = 0 (Theorem 1.2), and derives a corollary that the rank over Q(i) is also 0. The proof is based on an explicit 2-isogeny descent: six homogeneous spaces are tested for local solvability using a classification of quartic equations over Q_p imported from Cohen, and a cokernel C_S originating from the exact sequence is controlled to separate rank and Sha.

Significance. If the proof is completed, this is a substantial contribution to the arithmetic of one- and two-parameter families of elliptic curves. It extends Silverman's classical result for y^2 = x^3 + p x to the two-prime family and strengthens an earlier special case by the first author. The classification is genuinely derived, not fitted: the four congruence conditions are outputs of explicit local-solvability checks with no free parameters. The use of the cokernel C_S to separate rank and Sha is a useful idea that goes beyond the minimal-Selmer statement. The paper is mostly explicit and checkable, and the computational part of the argument appears correct.

major comments (1)

[Section 5 and Lemma 2.2] The converse of Theorem 1.2 depends on Lemma 2.2 to conclude that dim_2 C_S is even, and this is load-bearing in the case dim_2 S(φ̂)=3: an odd value of dim_2 C_S would allow rank + dim X[2] = 0 with non-minimal Selmer groups (e.g., dim S(φ)=2, dim S(φ̂)=3, dim C_S=3). The proof of Lemma 2.2, however, only cites Fisher's Theorem 2.1 and asserts, in two sentences, that Fisher's S_1, S'_1, S_2, S'_2 match the Selmer groups here. Fisher's hypotheses are not stated and the identifications are not verified for the specific curves E_{p,q}. Please provide a complete statement of Fisher's theorem, verify its hypotheses (including the rational 2-torsion point and the precise correspondence of Selmer groups), or give a self-contained proof of Lemma 2.2.

minor comments (7)

[Title] The title has a typo: 'CUR VES' should be 'CURVES'.
[Section 3.2, Remark after Table 2] The stated change of variables 'z → 2z' to remove a factor of 16 is in the wrong direction; one needs z → z/2. For example, bC_{-1} should be obtained from w^2 = -u^4 + 16pq z^4 by replacing z with z/2, not 2z.
[Section 3.1] The sentence 'only three of the equations C_d are actually distinct modulo squares' is imprecise: d=p and d=q are not equal modulo squares, but their associated homogeneous spaces are isomorphic via interchanging the variables. Please clarify this point, since the later reduction to C_2, C_p, C_{2p} relies on it.
[Notation] The notation dim_2 is nonstandard and visually confusing; consider writing dim_{F_2} or dim_{\mathbb{F}_2} throughout.
[Section 4] The case analysis is long and terse. A summary table or a short diagram showing which local conditions eliminate which homogeneous spaces would improve readability.
[Section 5] In the dim_2 S(φ̂)=3 case, the statement '-1, p ∈ S(φ̂)' follows because S(φ̂) is a 3-dimensional subspace of the 3-dimensional span of -1,p,q; this point should be made explicit.
[Section 3.1] It is stated without proof that d=1 and d=pq always lie in S(φ), and d=1 and d=-pq always lie in S(φ̂). This is standard, but a one-line justification would be helpful for the self-contained reader.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the classification is derived from explicit local-solvability checks; the sole self-citation is a non-load-bearing special-case benchmark.

full rationale

The paper's central claim (Theorem 1.2) is not circular. Theorem 1.1 is derived by applying Cohen's Proposition 6.5.1 (quoted as Lemma 3.1) to six explicit homogeneous spaces listed in Tables 1 and 2. The final congruence conditions (mod 8, mod 16, Legendre symbols) are outputs of these local-solvability computations, not inputs fitted to force the desired conclusion. There are no free parameters and no post-hoc exclusions: each Selmer group is either accepted or rejected according to the stated local criteria. Theorem 1.2's forward direction follows immediately from Theorem 1.1 and Proposition 2.1. Its converse uses Lemma 2.3 and Lemma 2.2; Lemma 2.2 is imported from Fisher's Theorem 2.1 (reference [6]), which is an external theorem, not a self-citation. Even though the paper identifies Fisher's S_1, S'_1, S'_2 with its Selmer groups in only two sentences and does not verify Fisher's hypotheses, that is a proof-verification concern, not circularity: it does not reduce the theorem to its own statement. The only self-citation is [7], used in Example 1.3 as a special case of the new result ('we obtain Theorem 1.1 in [7] as a special case of our result'). That citation is never used as an input to any proof and is not load-bearing. Proposition 2.1 is proved from Silverman's exact sequences and the snake lemma rather than assumed. No ansatz is smuggled in via citation: the homogeneous-space equations and local conditions are explicit and externally checkable. Thus the derivation is self-contained relative to its stated external lemmas, with no circular reduction exhibited.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper is a derivation, not a fit: Theorem 1.1's conditions are outputs of explicit local-solvability checks on six homogeneous spaces, and Proposition 2.1 is proved from Silverman's exact sequences. The unpaid inputs are two published external theorems - Cohen's Proposition 6.5.1 (Lemma 3.1) and Fisher's Theorem 2.1 (Lemma 2.2) - both legitimate but neither re-derived nor computationally cross-checked here, plus the standard Silverman descent machinery. C_S is the cokernel of a map between existing, well-defined groups, isomorphic by the snake lemma to a cokernel acting on Sha(E/Q)[2]; it is a defined object, not a postulated entity. No free parameters and no invented entities; the main verification burden for a reader is the four-way case analysis in Section 4 and the 2-adic branch of Lemma 3.1.

assumptions (7)

domain assumption Cohen, Proposition 6.5.1 (Lemma 3.1 here): complete local-solvability classification for a x^4 + b y^4 + c z^2 = 0 over R and Q_p, including the p = 2 replacement rules.
Quoted verbatim without proof; it is the engine for Lemmas 3.2-3.8 and hence for every mod-8/mod-16 and Legendre-symbol condition in Theorem 1.1. External published support, but load-bearing and un-re-derived.
domain assumption Fisher, Theorem 2.1: alternating pairings on the two isogeny Selmer groups with kernels S_2 and S'_2, giving a nondegenerate alternating pairing on S'_1/S'_2; consequently |C_S| is a square.
Used in Lemma 2.2 to conclude dim_2 C_S is even; that evenness is load-bearing in the converse of Theorem 1.2 (Section 5, dim S(phi-hat) = 3 case). The identification of the paper's groups with Fisher's S_1, S'_1, S'_2 is asserted in two sentences; Fisher's hypotheses are not restated or verified.
standard math Silverman X.4.2(a) and the page-336 exact sequence for a degree-2 isogeny (the 2-isogeny descent exact sequences).
Basis of Proposition 2.1: equations (2.1), (2.2), (2.8) and the exact sequence producing (2.5).
standard math Standard 2-descent exact sequence 0 -> E(Q)/2E(Q) -> S^(2)(E/Q) -> Sha(E/Q)[2] -> 0, and dim E(Q)/2E(Q) = rank + dim E(Q)[2].
Used in the proof of Proposition 2.1 and in Theorem 1.2 (with dim E(Q)[2] = 1 for E_{p,q}, from Silverman X.6.1(a)).
standard math Quadratic reciprocity and the supplementary laws for (2/p) and (-1/p).
Used throughout Section 4 to equate (p/q) with (q/p) in the p=1 (mod 4) cases and to flip signs in the p=q=3 (mod 4) case.
standard math Mordell-Weil finiteness and the basic finite-dimensional F_2-vector-space structure of Selmer-group quotients.
Background for all dimension counts in Proposition 2.1 and Lemmas 2.2-2.3.
standard math Proposition 6.1 (rank over a quadratic field equals rank over Q plus rank of the quadratic twist; Silverman Exercise 10.22(c)(iv)).
Used only for Corollary 1.5 and Corollary 6.2; not load-bearing for the main theorems.

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Pith. "Pith review of Exact classification of elliptic curves $y^{2}=x^{3}-pqx$ with rank $0$ and trivial $\Sha[2]$." pith.science (2026). https://pith.science/paper/3SAJF2A4

@misc{pith2026260714033,
  author       = {Pith},
  title        = {Pith review of: Exact classification of elliptic curves $y^2=x^3-pqx$ with rank $0$ and trivial $\Sha[2]$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SAJF2A4}},
  note         = {Machine review of arXiv:2607.14033}
}

abstract

For the elliptic curves $E_{p,q}: y^{2}=x^{3}-pqx$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which rank$\,E_{p,q}(\mathbb{Q})$ and $\dim_{\mathbb{F}_{2}} \Sha \left( E_{p,q}/\bbQ \right)[2]$ are both $0$. We do so via a similar characterisation of when the Selmer groups associated with the degree-$2$ isogeny $\phi$ and its dual $\widehat{\phi}$ are both of minimal size, along with results about a cokernel that arises from a related exact sequence.

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Works this paper leans on

9 extracted references · 2 linked inside Pith

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [2]

Bhargava and A

M. Bhargava and A. Shankar,Ternary cubic forms having bounded invariants and the existence of a positive proportion of elliptic curves having rank0, Ann. of Math.181(2015), 587–621

2015

[2] [1]

Bhargava and A

M. Bhargava and A. Shankar,Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math.181(2015), 191–242

2015

[3] [3]

Bhargava and A

M. Bhargava and A. Shankar,The average number of elements in the4-Selmer groups of elliptic curves is7,https://arxiv.org/abs/1312.7333

arXiv

[4] [4]

Bhargava and A

M. Bhargava and A. Shankar,The average size of the5-Selmer group of elliptic curves is6, and the average rank is less than1,https://arxiv.org/abs/1312.7859

arXiv

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Cohen,Number Theory Volume I: Tools and Diophantine Equations, Springer, 2007

H. Cohen,Number Theory Volume I: Tools and Diophantine Equations, Springer, 2007

2007

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Fisher,Higher descents on an elliptic curve with a rational2-torsion point, Math

T. Fisher,Higher descents on an elliptic curve with a rational2-torsion point, Math. Comp.86(2017), 2493–2518

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Ghosh,Rank of a certain family of elliptic curves, Monatsh

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J. H. Silverman,The Arithmetic of Elliptic Curves, 2nd ed., Springer, 2009

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