Torsion groups of Mordell curves over cubic and sextic fields

Bidisha Roy; Pallab Kanti Dey

REVIEW 3 major objections 4 minor 21 references

Torsion groups of Mordell curves over cubic and sextic fields

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper determines the complete list of torsion subgroups of Mordell curves y^2 = x^3 + c over cubic and sextic number fields, including exact conditions for each group.

desk verdict Solid new torsion classifications for cubic and rational sextic fields, but Theorem 4's exclusion of Z/10Z rests on a false divisibility claim in Lemma 7.2. read the letter →

arxiv 1908.07791 v1 pith:BKBLJLZL submitted 2019-08-21 math.NT

Pallab Kanti Dey , Bidisha Roy This is my paper · ORCID

classification math.NT MSC 11G0511R1611R2114H52

keywords EllipticcurvesTorsiongroupsNumberfieldsMordellCubicSexticComplexmultiplication

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 tries to settle a precise question: for a Mordell curve $y^2 = x^3 + c$, which finite abelian groups can appear as the torsion subgroup — the group of points of finite order — when one takes points over a cubic or a sextic number field? For Mordell curves the claimed answer is short: over a cubic field only $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z}$, and $\mathbb{Z}/9\mathbb{Z}$ (plus the trivial group); over a sextic field the same list plus five product groups and three exceptional groups $\mathbb{Z}/19\mathbb{Z}$, $\mathbb{Z}/7\mathbb{Z}$, and $\mathbb{Z}/14\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. In the rational-coefficient case the paper goes further and pins down exactly which group occurs, in terms of whether $c$ is a square or a cube in the field and whether certain fixed fields lie inside $K$. A complete classification of this family matters because the same question for arbitrary elliptic curves over fields of degree 3 or higher is still open, and Mordell curves are the natural test family among complex-multiplication curves.

What carries the argument

The machinery runs on two tracks. The first is division-polynomial analysis of the family $y^2 = x^3 + c$: a point of order 2 exists exactly when $c$ is a cube in $K$, a point of order 3 when $c$ is a square in $K$ or when $-3c$ is a square and $4c$ is a cube in $K$, and a point of order 9 forces $c$ to be a square, $4c$ a cube, and $K$ to contain the normal cubic field $\mathbb{Q}(r)$ defined by $r^3 - 3r^2 + 1 = 0$ — the same polynomial recurs in every degree. The second track is reduction modulo primes: for a prime $p \equiv 2 \pmod{3}$ of good reduction the point count on the reduced curve is $p^f + 1$ with $f = 1, 2, 3,$ or $6$, and the injectivity of the reduction map on torsion then caps the possible orders; in the sextic case a supersingularity criterion pins the trace of Frobenius to $\pm p^3$ or $\pm 2p^3$. The exceptional groups are built through the Kubert–Tate normal form, the equation $y^2 + (1-c)xy - by = x^3 - bx^2$ that every elliptic curve with a point of order at least 4 admits, with the $j$-invariant matched to the Mordell form.

What would settle it

Check the divisibility claims of Lemma 7.2 at $p = 1409$: since $1409 \equiv 5 \pmod{156}$ and $5 \mid 1409^3 + 1$, the lemma is refuted exactly as written, and the fate of Theorem 4 is then decided by whether some sextic field $K$ and coefficient $c \in K$ make $y^2 = x^3 + c$ carry a point of order 5 or 13 — a direct search over sextic fields containing $\omega$ would settle it.

Watch

Extended reading notes

Core claim

The central discovery is a classification theorem. For any Mordell curve $E : y^2 = x^3 + c$, the torsion subgroup over a cubic field is one of $\mathbb{Z}/9\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$, or the trivial group (Theorems 1 and 2). Over a sextic field it is one of those groups, one of the five product groups $\mathbb{Z}/9\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}/6\mathbb{Z}$, $\mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, or one of the three exceptional groups $\mathbb{Z}/19\mathbb{Z}$, $\mathbb{Z}/7\mathbb{Z}$, $\mathbb{Z}/14\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ (Theorems 3 and 4). When the curve is defined over $\mathbb{Q}$ with $c$ a sixth-power-free integer, the theorems attach to each group necessary and sufficient conditions expressed as elementary field statements: whether $c$ is a square or a cube in $K$, whether $4c$ is a cube, whether $-3c$ is a square, whether the cube root of unity $\omega$ belongs to $K$, and whether $K$ contains the normal cubic field $\mathbb{Q}(r)$ for $r$ satisfying $r^3 - 3r^2 + 1 = 0$. The three exceptional groups are realized by explicit sextic fields and explicit coefficients, so the list is claimed to be exhaustive as well as necessary.

Load-bearing premise

The exclusion of order-5 and order-13 torsion over sextic fields rests entirely on the claim that for every prime $p \equiv 5 \pmod{156}$, neither $5$ nor $13$ divides $p^3 \pm 1$ or $p^6 \pm p^3 - 1$; that claim is false as stated, since $p = 1409$ satisfies the congruence yet $5$ divides $p^3 + 1$.

Editorial extensions

If this is right

Over cubic fields the torsion of any Mordell curve is decided by two yes/no questions — whether $c$ is a square in $K$ and whether $c$ is a cube in $K$ — plus one exceptional case, so the classification is fully algorithmic.
For rational Mordell curves over sextic fields, all five groups beyond the cubic list require the cube root of unity $\omega$ to lie in $K$; if $\omega \notin K$, the torsion is already among the cubic-field possibilities.
The three exceptional groups, $\mathbb{Z}/19\mathbb{Z}$, $\mathbb{Z}/7\mathbb{Z}$, and $\mathbb{Z}/14\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, occur only for Mordell curves that are not defined over $\mathbb{Q}$, so they are genuinely new torsion phenomena rather than base changes of rational curves.
Order-9 torsion exists only inside fields that contain the normal cubic field defined by $r^3 - 3r^2 + 1 = 0$; deciding whether that polynomial has a root in $K$ is what separates order 3 from order 9.

Reading between the lines

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

The same square-and-cube bookkeeping should carry over to degree-4 fields, where the point-count bound runs over residual degrees 1, 2, and 4 and no exceptional cubic subfield has to be tracked; assembling the analogue of $\Phi^M(4)$ looks like a finite case check.
The fixed identity $r^3 - 3r^2 + 1 = 0$ governing order-9 torsion suggests that the whole classification could be restated Galois-theoretically: torsion growth is controlled by whether $K$ contains $\omega$ and this one normal cubic field, independent of the coefficient $c$ except through its square/cube class.
A direct computational test of Theorem 3 would enumerate sixth-power-free $c$ and sextic fields containing $\omega$ with small discriminant, then compare the predicted group with the computed torsion; the occurrence conditions are explicit enough to make this a finite search.

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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

Two things you should know. First, the paper actually does what it claims for cubic fields and for rational Mordell curves over sextic fields: Theorems 1–3 give complete lists with explicit appearance conditions, and those results are not in the prior literature. Second, Theorem 4 has a load-bearing arithmetic slip: Lemma 7.2 claims a non-divisibility statement that is false for the prime 5, so the submitted proof does not eliminate Z/10Z. The theorem may still be true, but the text as written does not prove it.

The new material is real. The sets Phi^M_Q(3), Phi^M(3), Phi^M_Q(6), and Phi^M(6) are not in the cited references; [3] explicitly left d=3 and d=6 open. The proof strategy is standard reduction and division-polynomial argument, and for Theorems 1–3 it is handled carefully. The appearance conditions in Theorems 1 and 3 are concrete and checkable, and the small-order torsion lemmas (2, 3, 4, 9, 18, 27) are mostly coherent.

The soft spots are concentrated in Section 7. Lemma 7.2 chooses primes p ≡ 5 mod 156 and then asserts that because p ≡ 5 mod 13 and ℓ is either 5 or 13, ℓ divides neither p^3 ± 1 nor p^6 ± p^3 − 1. The mod-13 condition only controls divisibility by 13; it says nothing about 5. Concretely, p = 1409 satisfies p ≡ 5 mod 156, yet p ≡ 4 mod 5, so 5 | p^3 + 1. The non-divisibility premise fails for ℓ=5, so the argument excluding Z/10Z collapses. The ℓ=13 part is unaffected by this particular mistake, but as written Lemma 7.2 does not establish the exclusion of both groups. Lemma 7.1's proof is omitted under hypotheses that differ from the cited similar lemma, and Lemma 7.6 relies on Magma computations without scripts or output, which makes the existence of the three exceptional groups harder to verify.

None of this undermines Theorems 1–3, and I expect the sextic-field picture is close to correct. But Theorem 4 is one of the two headline claims, and its proof has a concrete false assertion. The right fix is a different congruence or a finer local argument for the prime 5. That is a realistic repair, not a conceptual dead end.

Who this is for: people working on torsion classifications over number fields, especially those who care about CM families. I'd want to see a corrected version before citing it, but I would not desk-reject this. Send it to a referee, ask them to focus on Section 7, and expect a revision.

Referee Report

3 major / 4 minor

Summary. The paper studies torsion subgroups of Mordell curves E: y^2 = x^3 + c over cubic and sextic fields. Theorem 1 classifies Phi_M^Q(3) and gives explicit conditions for each possible group; Theorem 2 classifies Phi_M(3); Theorem 3 classifies Phi_M^Q(6) and gives conditions; Theorem 4 asserts that Phi_M(6) equals Phi_M^Q(6) together with the three exceptional groups Z/19Z, Z/7Z, and Z/14Z direct-sum Z/2Z. The proof strategy for the cubic cases combines elementary analyses of 2-, 3-, 4-, 6-, 9-, 18-, and 27-torsion with reduction maps and point counts over finite fields. The sextic case starts from Clark, Corn, Rice, and Stankewicz's list of CM torsion groups over sextic fields, then attempts to eliminate the non-occurring groups by congruence arguments and Magma computations, and to realize the remaining groups by explicit curves.

Significance. If the classification is correct, it completely determines the possible torsion structures for Mordell curves over cubic and sextic fields, with explicit realizability conditions. The paper has useful strengths: the elementary point-counting arguments for the cubic cases are mostly transparent, the criteria in Theorems 1 and 3 are explicit and checkable, and the paper provides concrete curves for at least some exceptional groups. The main weakness is concentrated in Theorem 4: Lemma 7.2 contains a false modular-arithmetic assertion that invalidates the exclusion of Z/10Z as written, Lemma 7.1 is stated without a proof even though it is not a formal consequence of the cited lemma, and several load-bearing claims are delegated to unstated Magma computations. These issues are localized and plausibly repairable, but they currently prevent the paper from establishing its central sextic-field claim.

major comments (3)

[Lemma 7.2] The final paragraph of Lemma 7.2 asserts that because p is congruent to 5 modulo 13 and l is either 5 or 13, neither 5 nor 13 divides p^3 +/- 1 or p^6 +/- p^3 - 1. This is false for l=5: the congruence p congruent to 5 modulo 13 controls divisibility by 13 only, and says nothing about divisibility by 5. For example, p=1409 satisfies p congruent to 5 modulo 156, hence p congruent to 5 modulo 13, but p is congruent to 4 modulo 5, so p^3+1 is congruent to 4^3+1, which is 0 modulo 5. Since Z/10Z appears in the Clark et al. list in equation (9) and is not in Phi_M^Q(6), the elimination of Z/10Z in Theorem 4 rests entirely on this false non-divisibility assertion. This is load-bearing for the equality in Theorem 4. The argument is likely repairable by choosing p in a residue class modulo 5 that makes the displayed values nonzero, for example p congruent to 2 modulo 5, and then using Dirichlet with the combined modulus; the authors should supply the corrected congruence argument.
[Lemma 7.1] The proof of Lemma 7.1 is omitted with the words "By the similar approach of the proof of Lemma 6.4", but Lemma 6.4 is proved in Section 6 under the standing assumption that E is a rational Mordell curve, namely c in Q, whereas Lemma 7.1 must exclude 4-torsion for every c in K. The key step in Lemma 6.4 passes from x^3 = (-10 +/- 6 sqrt(3)) a^3 to x = (-1 +/- sqrt(3)) a; this requires knowing that the relevant cube root of unity lies in K, and that condition is not automatic when c is allowed to vary in K. Thus the exclusions of Z/4Z and Z/2Z direct-sum Z/4Z are not supported as written. Please provide a complete proof or a precise reference whose hypotheses match the statement of Lemma 7.1.
[Lemma 7.6] Several load-bearing computational claims are asserted as "by using magma" without commands, outputs, or enough data for independent verification. This affects the irreducibility and torsion-group assertions in Lemma 4.6, Lemma 4.7, Lemma 5.8, Lemma 7.3, and in particular Cases 2 and 3 of Lemma 7.6. For the existence half of Theorem 4, the claims that E(K)_tors is isomorphic to Z/7Z and to Z/2Z direct-sum Z/14Z need reproducible verification. Additionally, for the Kubert-Tate examples in Lemma 7.6, the statement that equation (2) shows the displayed curve is a Mordell curve is not demonstrated; one needs to check that the j-invariant is 0. Please include the relevant code and outputs, or give a proof, and verify the j=0 condition explicitly.

minor comments (4)

[Lemma 7.2] The possible group orders for residue degree f_i=6 are p^6 +/- p^3 + 1 or (p^3 +/- 1)^2, not p^6 +/- p^3 - 1 as written; the sign error should be corrected.
[Section 7, end] The line "Proof of Theorem 2" at the end of Section 7 should read "Proof of Theorem 4".
[Proof of Theorem 3, Case 1(b)] The notation "4^{1/3} in K" should be replaced by the precise condition "4 is a cube in K" or equivalently "4c is a cube in K" given that c is a cube.
[Lemma 4.1] In the ideal decomposition pO_K = P_1^{e_1} ... the text asserts 0 <= e_i <= 1, but ramified primes in a cubic field can have e_i = 2 or 3; the later conclusion still appears to go through, but the statement should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the torsion classifications are derived from standard point-count lemmas and external CM torsion results, with only minor author self-citations that are not load-bearing.

full rationale

The derivation chain is not circular. The main target is the classification of torsion subgroups of Mordell curves y^2 = x^3 + c over cubic and sextic fields. Theorem 4 starts from the external Clark-Corn-Rice-Stankewicz list of all torsion subgroups of CM elliptic curves over sextic fields ([1], displayed as (9)), restricts to Mordell curves, and then eliminates groups via reduction point counts, division polynomials, and magma checks; the existence examples are given by explicit Kubert-Tate normal forms whose j-invariant is 0. The only self-citations are Lemma 3.2, attributed to Corollary 1 of the first author's earlier paper [3], and Proposition 3, attributed to [4]. Lemma 3.2 is a standard point count for E:y^2=x^3+c at primes p≡2 mod3 and is stated with hypotheses that do not in any way assume the paper's torsion classification; Proposition 3 is the standard injectivity of reduction on torsion away from bad primes. These are independent, parameter-free facts that can be checked directly, so they are real evidence rather than circular imports. No fitted parameter is later relabeled as a prediction, and no uniqueness theorem from the authors' own prior work is invoked to force the classification. The paper does contain serious correctness defects that are not circularity: in Lemma 7.2 the assertion that for p≡5 mod156 and l=5 or 13 neither l divides p^3±1 nor p^6±p^3−1 fails for l=5 (e.g., p=1409, since 1409≡5 mod156 but 1409≡4 mod5 gives p^3+1≡0 mod5), so the proof as written does not exclude Z/10Z. That is a mathematical error, as are the omitted proof of Lemma 7.1 and unverified magma claims, but none of these amount to the paper deriving its conclusion from that same conclusion by definition. The appropriate circularity score is therefore low, reflecting the minor presence of author self-citations rather than any self-referential derivation.

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

No free parameters or invented entities appear. The central claim rests on standard theorems, a cited point-count lemma, a previously published CM torsion list, and unverifiable Magma checks.

assumptions (6)

standard math Dirichlet's theorem on primes in arithmetic progressions
Invoked in Lemmas 4.1, 5.1, 6.1 and 7.2 to produce primes p congruent to 2 mod 3q with good reduction.
domain assumption Point-count formula for y^2 = x^3 + c over F_{p^n} when p is congruent to 2 mod 3
Lemma 3.2 cited from [3], a previous paper by the first author, and used throughout; it is a standard CM curve count but is not re-derived here.
domain assumption Clark et al. list of possible torsion groups of CM elliptic curves over sextic fields, displayed in (9)
Used as the starting upper bound for Theorem 4; not proved in this paper.
standard math Quadratic-twist torsion decomposition E(K(sqrt(d)))[n] isomorphic to E(K)[n] x E_d(K)[n]
Lemma 3.1, used in Lemma 6.6 to bound 27-torsion.
standard math Injectivity of the reduction map on torsion outside finitely many primes
Proposition 3, used in every prime-reduction argument to transfer torsion from the number field into finite field groups.
ad hoc to paper Magma irreducibility and group-order computations
Several lemmas, including 4.6, 5.8, 7.3 and 7.6, rely on unstated Magma computations; no code, version, or certificates are provided.

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Cite this review

Pith. "Pith review of Torsion groups of Mordell curves over cubic and sextic fields." pith.science (2026). https://pith.science/paper/BKBLJLZL

@misc{pith2026190807791,
  author       = {Pith},
  title        = {Pith review of: Torsion groups of Mordell curves over cubic and sextic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BKBLJLZL}},
  note         = {Machine review of arXiv:1908.07791}
}

read the original abstract

In this paper, we classify torsion groups of rational Mordell curves explicitly over cubic fields as well as over sextic fields. Also, we classify torsion groups of Mordell curves over cubic fields and for Mordell curves over sextic fields, we produce all possible torsion groups.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

P. K. Dey, Torsion groups of a family of elliptic curves over number ﬁel ds, Czech. Math. Journ. (2018)

work page 2018

[2] [1]

P. L. Clark, P. Corn, A. Rice and J. Stankewicz, Computation on elliptic curves with complex multiplicatio n, LMS J. Comput. Math. 17 (1) (2014), 509-535

work page 2014

[3] [2]

H. B. Daniels and E. Gonz´ alez-Jim´ enez, On the torsion of rational elliptic curves over sextic ﬁelds , arXiv:1808.02887v1. (2018)

work page Pith review arXiv 2018

[4] [4]

P. K. Dey, Elliptic curves with rank 0 over number ﬁelds, Funct. Approx. Comment. Math. 56 (1) (2017), 25-37

work page 2017

[5] [5]

G. Fung, H. Str¨ oher, H. Williams, and H. Zimmer. Torsion groups of elliptic curves with integral j-invariant over pure cubic ﬁelds. J. Number Theory 36 (1990) 1245. 2

work page 1990

[6] [6]

Gonz´ alez-Jim´ enez,Complete classiﬁcation of the torsion structures of ration al elliptic curves over quintic number ﬁelds, J

E. Gonz´ alez-Jim´ enez,Complete classiﬁcation of the torsion structures of ration al elliptic curves over quintic number ﬁelds, J. Algebra. 478 (2017), 484-505

work page 2017

[7] [7]

Growth of torsion groups of elliptic curves upon base change

E. Gonz´ alez-Jim´ enez and F. Najman, Growth of torsion groups of elliptic curves upon base change . arXiv:1609.02515v6. (2019)

work page Pith review arXiv 2019

[8] [8]

Gonz´ alez-Jim´ enez and J

E. Gonz´ alez-Jim´ enez and J. M. Tornero,Torsion of rational elliptic curves over quadratic ﬁelds, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM 108 (2) (2014 ), 923-934

work page 2014