{"id":"2e4056e7-7d55-4477-a622-8f6e81c9a052","arxiv_id":"1908.07791","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For Mordell curves over cubic fields the torsion subgroup is one of five cyclic groups; over sextic fields the rational-coefficient list gains five product groups, and arbitrary sextic coefficients add Z/19, Z/7 and Z/14 + Z/2.","lead":"This paper classifies the possible torsion groups of Mordell curves, elliptic curves of the form y2 = x3 + c, after extending scalars to cubic and sextic number fields. It gives the full list of groups and exact conditions on c and the field under which each occurs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7.2's divisibility claim is false for l=5: p=1409 satisfies p≡5 mod156 yet 5 | p^3+1, so the proof of Theorem 4 does not exclude Z/10Z.","rationale":"The reader's weakest_assumption exactly identifies the load-bearing error. I independently verified p=1409: 1409=156·9+5, 1409 mod 5 = 4, so p^3+1 ≡ 4^3+1 = 65 ≡ 0 (mod 5). The proof of Lemma 7.2 therefore does not establish the exclusion of Z/10Z, which is essential for Theorem 4. No other independent check of the paper changes this assessment; the omission of Magma verification for Lemma 7.6 constructions is a secondary concern but not needed for the verdict. Thus the reader's REJECT is appropriate, and my read does not change the verdict.","tokens_in":17820,"tokens_out":9213,"duration_ms":85153,"concrete_test":"Compute p^3+1 modulo 5 for p=1409 (the smallest prime congruent to 5 mod 156 that is 4 mod 5): the result is 0, contradicting the lemma's universal assertion. This settles that the written proof of Theorem 4 fails to exclude Z/10Z; whether the theorem can be repaired would require finding a new argument or finding an actual curve with 10-torsion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 7.2 the authors choose primes p ≡ 5 (mod 12q) with q=13 and then assert that because p ≡ 5 (mod 13) and l is either 5 or 13, neither 5 nor 13 divides p^3±1 or p^6±p^3−1. The mod-13 congruence controls divisibility by 13 only; it says nothing about divisibility by 5. Indeed p=1409=9·156+5 satisfies p ≡ 5 (mod 156), hence p ≡ 5 (mod 13), but p ≡ 4 (mod 5), so p^3+1 ≡ 4^3+1 ≡ 0 (mod 5). Thus the premise of Lemma 7.2 fails for l=5. Since Z/10Z appears in the Clark-Corn-Rice-Stankewicz list (9) and is not in Phi^M_Q(6), its elimination in Theorem 4 rests entirely on this false non-divisibility assertion. The l=13 part still works, so Z/26Z is not affected by this particular error, but the proof as written leaves Z/10Z unexcluded. This is load-bearing: Theorem 4's equality could be false if a Mordell curve over a sextic field has 10-torsion, and at minimum the submitted argument does not rule it out. The companion omissions (Lemma 7.1 proof, unverified Magma computations in Lemma 7.6) compound the issue, but the arithmetic error is the decisive defect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1585,"tokens_out":1819,"duration_ms":723541,"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":[{"comment":"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.","section":"Lemma 7.2"},{"comment":"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.","section":"Lemma 7.1"},{"comment":"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.","section":"Lemma 7.6"}],"minor_comments":[{"comment":"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":"Lemma 7.2"},{"comment":"The line \"Proof of Theorem 2\" at the end of Section 7 should read \"Proof of Theorem 4\".","section":"Section 7, end"},{"comment":"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.","section":"Proof of Theorem 3, Case 1(b)"},{"comment":"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.","section":"Lemma 4.1"}],"recommendation":"major_revision","confidential_remarks":"The false assertion in Lemma 7.2 is a serious technical error, but it appears localized and repairable by adding a congruence condition modulo 5, so I recommend major revision rather than rejection. Before acceptance, the authors should also be required to make the Magma computations reproducible and to give a complete proof of Lemma 7.1, since all three issues affect the central Theorem 4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nNone 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.\n\nWho 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.","headline":"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.","tokens_in":18685,"tokens_out":3552,"would_cite":false,"duration_ms":28893,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11R16","11R21","14H52"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Elliptic curves","Torsion groups","Number fields","Mordell curves","Cubic fields","Sextic fields","Complex multiplication"],"falsifier":"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.","tokens_in":17561,"feed_emoji":"🔢","tokens_out":26169,"duration_ms":195261,"temperature":0.7,"pith_summary":"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.","feed_headline":"All torsion groups of Mordell curves in degrees 3 and 6 are classified","feed_subtitle":"Every possible torsion group is named, with exact conditions for each; sextic fields add three new ones.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the full list of torsion structures for complex-multiplication elliptic curves over sextic fields, the starting set that Theorem 4 prunes, and the explicit sextic field and curve with Z/19Z torsion.","marker":"[1]"},{"why":"Supplies the point-count lemma for Mordell curves over finite fields used in all order-exclusion arguments, and the earlier determination of Phi^M_Q(d) for d = 2 and d >= 5 coprime to 6.","marker":"[3]"},{"why":"Supplies Proposition 3, the injectivity of the reduction map on torsion subgroups, which every exclusion lemma invokes.","marker":"[4]"},{"why":"Supplies Lemma 3.1 on torsion growth in quadratic twists, used to rule out order-27 torsion over sextic fields.","marker":"[8]"},{"why":"Supplies the Kubert–Tate normal form used to construct the exceptional curves carrying Z/19Z and Z/7Z torsion.","marker":"[13]"},{"why":"Supplies the supersingularity criterion and the trace congruence (Lemma 3.4) used in Lemma 7.2 to restrict the Frobenius trace in the sextic-field argument.","marker":"[20]"},{"why":"Supplies Proposition 1 on admissible trace values for elliptic curves over finite fields, used in Lemma 7.2 to pin a = ±p^3 or ±2p^3.","marker":"[21]"}],"fun_headline_variants":["Mordell curve torsion fully classified for cubic and sextic fields","Exact torsion groups of Mordell curves stated for degrees 3 and 6","Complete classification: torsion of Mordell curves over cubic and sextic fields","Mordell curves: all torsion groups in cubic and sextic fields listed","Torsion groups of Mordell curves precisely determined for degrees 3 and 6"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $p^3 + 1$.","fun_headline_variants_meta":{"raw":{"variants":["Mordell curve torsion fully classified for cubic and sextic fields","Exact torsion groups of Mordell curves stated for degrees 3 and 6","Complete classification: torsion of Mordell curves over cubic and sextic fields","Mordell curves: all torsion groups in cubic and sextic fields listed","Torsion groups of Mordell curves precisely determined for degrees 3 and 6"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000452,"raw_usage":{"total_tokens":2283,"prompt_tokens":959,"completion_tokens":1324,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":1237}},"tokens_in":575,"tokens_out":1324,"duration_ms":9403,"temperature":1.0,"reasoning_tokens":1237,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:58:39.321813+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the full list of torsion structures for complex-multiplication elliptic curves over sextic fields, the starting set that Theorem 4 prunes, and the explicit sextic field and curve with Z/19Z torsion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the point-count lemma for Mordell curves over finite fields used in all order-exclusion arguments, and the earlier determination of Phi^M_Q(d) for d = 2 and d >= 5 coprime to 6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 3, the injectivity of the reduction map on torsion subgroups, which every exclusion lemma invokes."},{"cited_title":"Gonz´ alez-Jim´ enez and J","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.1 on torsion growth in quadratic twists, used to rule out order-27 torsion over sextic fields."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Kubert–Tate normal form used to construct the exceptional curves carrying Z/19Z and Z/7Z torsion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the supersingularity criterion and the trace congruence (Lemma 3.4) used in Lemma 7.2 to restrict the Frobenius trace in the sextic-field argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 1 on admissible trace values for elliptic curves over finite fields, used in Lemma 7.2 to pin a = ±p^3 or ±2p^3."}],"review_version":1}