{"id":"f44b6cc5-55a8-426c-91ae-774f9778984a","arxiv_id":"1909.00637","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For rational CM elliptic curves, the torsion growth over quadratic fields is fully described by the CM invariants (cm, k): the possible growth groups are C3, C4, C6, C2 x C2, C2 x C4, C2 x C6 and C3 x C3, and the fields realizing each growth are listed explicitly.","lead":"This paper classifies how the torsion subgroups of rational elliptic curves with complex multiplication grow when the base field is extended to a quadratic field, and gives the explicit quadratic fields in terms of two invariants of the curve. This provides a complete, ready-to-use answer for an important low-degree case in a long-standing classification program.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the case analysis is internally consistent, and the main residual risk is inherited from published classifications.","rationale":"The paper's central claim, Theorem 3, is a complete table of torsion growth for rational CM elliptic curves over quadratic fields, indexed by the CM-invariants (cm,k). The proof is a case analysis over the thirteen CM classes using division polynomials and the published classifications of Bourdon-Clark-Stankewicz and Gonzalez-Jimenez-Tornero. In my review I focused on whether any internal step in this case analysis could be wrong or circular. The apparent ordering in Section 4.2, where the proof of Theorem 3 refers to the set Phi_CM_Q(2,G) from Table 1, is not a fatal circularity because the case analysis in each cm family independently establishes the exclusions by factoring division polynomials and checking square conditions; the set is used as an organizing device. I checked the cm=3 and cm=4 rows in detail because they involve sextic and quartic twist classes, and the conditions involving r are well-defined on the quotient Q*/(Q*)^n. The finite checks delegated to Magma are plausible and consistent with the displayed factorizations. The only substantive caveat is the completeness of the external group lists, which the reader also identified; however, this is a standard reliance on published theorems rather than a flaw in the paper's own argument. Thus the ACCEPT verdict should remain unchanged.","tokens_in":13847,"tokens_out":47625,"duration_ms":479424,"concrete_test":"Recompute all rows of Table 2 with an independent Magma or Sage script: for each cm in Table 3 and each class of k in Q*/(Q*)^n, factor the primitive division polynomials Psi_2, Psi_3, Psi_4 into irreducible factors over Q; for each linear or quadratic factor, test whether f(alpha) is a d-square in the corresponding quadratic field for some d in Q*; assemble the resulting set of quadratic growth fields and compare with Table 2. This directly verifies the finite case analysis, including the cm=3 and cm=4 rows where k is defined modulo 6th and 4th powers respectively.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. I checked the division-polynomial arguments in Section 4.2 family by family, and the entries of Table 2 are internally consistent: the stated factorizations, square conditions, and field computations support both the positive growth entries and the negative exclusions (e.g., no C2xC4 from G=C2, no C3xC3 from G=C1). The main residual risk is inherited rather than internal: the completeness of the possible-group lists in Table 2 relies on the published classifications [2, Theorem 1.4], [6], and [21, Theorem 2], and a few finite checks are delegated to Magma. These are substantial external results, and the author explicitly corrects a typo in [2], so the dependency is a caveat about completeness rather than a flaw in the paper's own derivation. Minor presentational issues, such as a notational slip in the cm=3 proof ('if k != -3' where the table and context require 'if r != -3'), do not affect the theorem or Table 2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper classifies the torsion growth of rational elliptic curves with complex multiplication (CM) after base change to quadratic fields. Theorem 1 determines the full set of possible torsion groups over quadratic fields, Theorem 2 determines, for each rational torsion group G, the possible growths and primitive torsion configurations, and Theorem 3 provides an explicit table depending only on the CM invariants (cm, k) that lists the quadratic fields over which torsion grows and the resulting groups. The proofs use division polynomials, the classification of CM torsion over quadratic fields by Bourdon, Clark, and Stankewicz, and the classification of torsion growth for rational curves by González-Jiménez and Tornero. The paper also corrects a typo in a published classification and states that all computations are reproducible with the provided Magma code.","tokens_in":14045,"tokens_out":41363,"duration_ms":353168,"significance":"If the proof gap noted below is repaired, the paper gives a complete and explicit answer to Problem 1 of González-Jiménez and Tornero in the CM case over quadratic fields: from two invariants one reads all growth fields with no search. This is a useful contribution to the program of understanding torsion growth over number fields. The paper also has the strengths of an explicit table, a case-based proof, machine-checked computations, and a careful correction of an earlier classification typo. The main caveat is that the completeness of Table 2 depends on previously published classifications, which are cited but not reproved; this is an inherited limitation rather than an internal flaw.","major_comments":[{"comment":"The proof that there are no points of order 4 over quadratic fields contains a false assertion. The text states: 'z = (−3 ± 2√2)k are the roots of the polynomial g(√x), but z ≠ x² for any x ∈ Q(√2) and k ∈ Q.' This is false: for k = −2, z = (−3 + 2√2)(−2) = 6 − 4√2 = (2 − √2)², so z is a square in Q(√2). The subsequent discussion of the roots ±√k concerns the order-4 points whose double is the rational 2-torsion point (0,0), not the roots of g, so it does not repair the gap. Since this is the only step excluding C4 in this case, the proof as written does not establish the corresponding row of Table 2. The conclusion appears correct and can be recovered by noting that if z is a square in Q(√2), then −k is twice a rational square, so the full 2-torsion field is Q(√2) and the presence of an order-4 point would force C2×C4, which is excluded by Table 1; alternatively one can check the y-coordinate square class. The manuscript should be revised to replace the false statement with a valid argument.","section":null}],"minor_comments":[{"comment":"The proof begins with 'Let H ∈ Φ_CM(2) \\ {C5, C7}'; this should be 'Φ_CM(2) \\ {C7, C10}', since C5 is not in Φ_CM(2) and the intended excluded groups are C7 and C10.","section":null},{"comment":"In the sentence 'We conclude that there are torsion growth to C2×C2 and C6 if k ≠ −3; and C2×C6 if r = −3', the condition 'if k ≠ −3' should read 'if r ≠ −3', since here k = r³ and the table distinguishes the value r = −3 (i.e., k = −27).","section":null},{"comment":"The exclusion of C4 for cm = 8 is stated too tersely: 'Since Q(√Δ(E8)) = Q(√2) we obtain that there are no points of order 4.' This conclusion does not follow from the equality of fields alone; it also uses the classification fact, already invoked in the cm = 7 case, that C2×C4 is not an admissible torsion group for a curve with rational torsion C2. Please make this argument explicit.","section":null},{"comment":"Several finite checks are asserted rather than displayed, for example 'We check that E²¹⁶₃(Q(√−3))tors ≃ C3×C3', the analogous statement for E⁻⁴³²₃, and the cm = 27 check. Since the Magma transcript is referenced, this is acceptable, but adding brief verification details or explicit references to the transcript would improve the exposition.","section":null}],"recommendation":"major_revision","confidential_remarks":"The false assertion in the cm = 4 proof is real but local, and the conclusion can be rescued by the classification argument already present in the cm = 7 case. I do not see a reason to doubt the theorem itself. The paper is within scope and the external classifications are standard, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, complete answer to a narrow natural problem. The genuinely new piece is Theorem 3, which tells you not just which torsion growth groups can occur for a rational CM curve over a quadratic field, but exactly which quadratic field realizes each growth, read off from the CM-invariants (cm, k). That is more than a re-parameterization of existing classifications; previous work had the possible groups, not the field-by-field description in invariant terms. The proof is a case analysis over the thirteen CM families using division polynomials, with careful use of published classification results. The author flags a typo in [2] and corrects it, which is a good sign.\n\nThe soft spots are real but minor. The completeness of Table 2 depends on external results, especially Bourdon–Clark–Stankewicz [2] and the Phi_Q(2,G) lists from the author's own earlier work. That is an inherited caveat, not an internal flaw; the paper explicitly leans on them and does not reprove completeness. A few finite checks are asserted rather than demonstrated, such as the check that E16_3(Q(sqrt(-3)))_tors is C3 x C3, though the Magma code is linked, so a referee can verify. There is a small notational slip in the cm=3 case, where the text says 'if k != -3' but the context requires 'r != -3'; it does not affect the theorem or Table 2.\n\nThe reader's take and the stress-test note both land: no load-bearing flaw surfaced. The paper is what it claims to be, a complete explicit classification for a natural family. It won't change the course of arithmetic geometry, but it sharpens the torsion-growth program and provides a useful test case for algorithmic approaches.\n\nFor a referee: the paper deserves serious peer review. The proofs are checkable, the dependency on external classifications is clearly declared, and the main result is a real addition. I would recommend acceptance after minor corrections.","headline":"Solid explicit classification of quadratic torsion growth for rational CM curves; Theorem 3 is genuinely new and the proof is checkable, with only minor inherited caveats.","tokens_in":14516,"tokens_out":1722,"would_cite":true,"duration_ms":36030,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For elliptic curves over Q with complex multiplication, this paper proves that torsion growth over quadratic fields is completely determined by two curve invariants, k and cm, and gives an explicit table of the fields where growth occurs.","keywords":["elliptic curves","complex multiplication","torsion subgroup","quadratic fields","CM-invariants","torsion growth","division polynomials","base change"],"falsifier":"Take E: y² = x³ − 2835x − 71442, the cm = 7 model. Theorem 3 predicts that over any quadratic field its torsion grows only to C2×C2 over Q(√−7); computing E(Q(√d))tors for all squarefree d and finding a single d with a point of order 4 would refute the table. Similarly, any rational CM curve that acquires a point of order 7 or 10 over a quadratic field would refute Theorem 1.","tokens_in":13659,"feed_emoji":"🔢","tokens_out":13579,"duration_ms":123606,"temperature":0.7,"pith_summary":"For an elliptic curve defined over Q with complex multiplication, this paper answers a structural question: over which quadratic fields does the torsion subgroup grow, and to which groups? The answer is a complete classification, expressed through two integers attached to the curve: the CM discriminant cm and the twisting parameter k. For every admissible pair (k, cm), Table 2 lists the base torsion group and, when growth occurs, both the larger torsion group and the explicit quadratic field Q(√d) that realizes it. The paper also determines the full set of torsion groups that can appear over quadratic fields for such curves (all torsion groups possible for CM elliptic curves over quadratic fields except C7 and C10) and shows that the maximum number of distinct growth fields is three. The result matters because it turns a computational search for growth into a lookup determined by invariants of the curve itself.","feed_headline":"One table maps CM curves to every quadratic torsion jump","feed_subtitle":"For rational CM elliptic curves, invariants k and cm fix every quadratic field where torsion grows.","key_machinery":"The carrier of the argument is the pair of CM-invariants (k, cm): cm is the absolute value of the discriminant of the quadratic order of complex multiplication, and k is the twisting parameter in Q*/(Q*)^{n(E)}, so every rational CM curve is Q-isomorphic to one of thirteen models E^k_cm. The proof detects torsion and its growth through the primitive 2-, 3-, and 4-division polynomials Ψ_n(x), whose roots are exactly the x-coordinates of points of exact order n, together with the identity E(Q(√d))[n] ≃ E(Q)[n] ⊕ E[d](Q)[n] for odd n, which identifies new odd-order torsion over a quadratic field with torsion of the corresponding twist over Q. Full 2-torsion is read from the discriminant, giving the field Q(√Δ(E_cm)). This mechanism turns the growth question into a finite check of whether certain values f_cm(α) are squares in the relevant quadratic field, and the outputs are organized by the (k, cm) invariants in Table 2.","core_discovery":"The central result, Theorem 3, states that if E/Q has CM and (cm, k) are its CM-invariants, then the torsion growth of E over quadratic fields depends only on these two integers. Table 2 lists, for each of the thirteen CM classes, exactly which groups Hi occur and the corresponding fields Q(√di) with E(Q(√di))tors ≃ Hi. This is complemented by two classification statements: Theorem 1 identifies the possible torsion groups over quadratic fields as the set Φ_CM(2) (the groups realized by CM elliptic curves over quadratic fields) minus the cyclic groups C7 and C10, and Theorem 2 gives the complete growth configurations for each possible base group G, showing that no curve has more than three distinct primitive growth fields. The proof proceeds case by case through the thirteen CM classes using division polynomials and a twist decomposition, and it corrects a typo in the prior quadratic-field classification it relies on.","pith_inferences":["The same (k, cm) parametrization is the natural template for the announced higher-degree growth tables; the quadratic case supplies the base case against which those tables can be checked.","The proof's reliance on square-class conditions on k suggests an elementary reformulation: for a fixed cm, the growth question becomes a statement about which squarefree integers k produce rational roots of low-degree division polynomials, independent of the ambient classification theorems.","The non-base-change exceptional cases in the quadratic classification—C7, C10, and some C2×C4 instances—are exactly the cases Theorem 3 separates out, so a conjectural invariant-based characterization of CM curves defined directly over quadratic fields could start from that separation."],"forward_implications":["For any rational CM curve, deciding whether torsion grows over a quadratic field—and to which group—becomes a matter of reading its (k, cm) entry in Table 2; no per-field computation is needed.","The groups C7 and C10 are impossible over quadratic fields for rational CM curves, although both occur for CM curves that are not base changes from Q; this is because 5- and 7-torsion never occur over Q, and odd-prime torsion cannot appear over a quadratic field if it is absent at the base.","For each starting torsion group, the complete list of growth configurations is known, and no curve can exhibit more than three distinct fields of primitive growth.","The listed fields are minimal in the sense of primitive growth: any quadratic field where growth occurs is exactly one of the Q(√di) entries, so growth over larger fields is a corollary of the table."],"supporting_citations":[{"why":"Supplies the classification of torsion of CM elliptic curves over quadratic fields that fixes the possible growth groups and identifies which exceptional fields occur; the paper corrects a typo in this result.","marker":"[2]"},{"why":"Supplies Φ_CM(2), the full list of torsion groups over quadratic fields, which Theorem 1 uses by removing C7 and C10.","marker":"[6]"},{"why":"Supplies Φ_Q(2,G), the possible torsion groups over quadratic fields for rational elliptic curves by base group, used to prune the growth list for each G.","marker":"[21]"},{"why":"Classifies torsion of CM elliptic curves over Q, giving the starting groups and the absence of 5- and 7-torsion used to rule out C7 and C10.","marker":"[30]"},{"why":"Provides the classical list of the thirteen CM orders and j-invariants on which the CM-invariant parametrization (cm, k) rests.","marker":"[33]"}],"fun_headline_variants":["Torsion growth of CM curves over quadratics fully classified","Explicit quadratic torsion map for rational CM elliptic curves","Invariants k and cm determine all quadratic torsion jumps","Complete table: CM elliptic torsion growth over quadratics","Thirteen CM classes, every quadratic torsion jump listed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the previously published classifications of torsion of CM elliptic curves over quadratic fields, and of rational elliptic curves over quadratic fields, are complete; the paper checks only the groups those lists declare possible, so a single missing group would make Table 2 incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Torsion growth of CM curves over quadratics fully classified","Explicit quadratic torsion map for rational CM elliptic curves","Invariants k and cm determine all quadratic torsion jumps","Complete table: CM elliptic torsion growth over quadratics","Thirteen CM classes, every quadratic torsion jump listed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000487,"raw_usage":{"total_tokens":2325,"prompt_tokens":794,"completion_tokens":1531,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":1449}},"tokens_in":410,"tokens_out":1531,"duration_ms":10854,"temperature":1.0,"reasoning_tokens":1449,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:41:42.324557+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take E: y² = x³ − 2835x − 71442, the cm = 7 model. Theorem 3 predicts that over any quadratic field its torsion grows only to C2×C2 over Q(√−7); computing E(Q(√d))tors for all squarefree d and finding a single d with a point of order 4 would refute the table. Similarly, any rational CM curve that acquires a point of order 7 or 10 over a quadratic field would refute Theorem 1.","supporting_citations":[{"cited_title":"Bourdon, P","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of torsion of CM elliptic curves over quadratic fields that fixes the possible growth groups and identifies which exceptional fields occur; the paper corrects a typo in this result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Φ_CM(2), the full list of torsion groups over quadratic fields, which Theorem 1 uses by removing C7 and C10."},{"cited_title":"Gonz´ alez-Jim´ enez, and J.M","cited_arxiv_id":null,"evidence_quote":"Supplies Φ_Q(2,G), the possible torsion groups over quadratic fields for rational elliptic curves by base group, used to prune the growth list for each G."},{"cited_title":"Olson, Points of finite order on elliptic curves with complex multip lication","cited_arxiv_id":null,"evidence_quote":"Classifies torsion of CM elliptic curves over Q, giving the starting groups and the absence of 5- and 7-torsion used to rule out C7 and C10."},{"cited_title":"Silverman, Advanced topics in the arithmetic of elliptic curves","cited_arxiv_id":null,"evidence_quote":"Provides the classical list of the thirteen CM orders and j-invariants on which the CM-invariant parametrization (cm, k) rests."}],"review_version":1}