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Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1909.00637 v2 pith:RDWWNFZ5 submitted 2019-09-02 math.NT math.AG

Enrique González-Jiménez This is my paper · ORCID
classification math.NTmath.AG MSC 11G0511G15
keywords ellipticcurvescomplexmultiplicationtorsionsubgroupquadraticfieldsCM-invariantsgrowthdivisionpolynomialsbasechange
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

1 major / 4 minor

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.

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 (1)
  1. 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.
minor comments (4)
  1. 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.
  2. 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).
  3. 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.
  4. 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Table 2 is derived by direct division-polynomial computations from independently published input classifications.

full rationale

Walking the derivation chain, Theorem 3's Table 2 is obtained by fixing each CM-invariant pair (cm,k), restricting the set of possible torsion-growth groups using published classifications ([2, Theorem 1.4], [6], and [21, Theorem 2]), and then checking the 2-, 3-, and 4-division polynomials to decide exactly which quadratic fields realize those groups. No fitted parameter is later renamed as a prediction: the CM-invariants are defined in Section 2.4 before any growth statement, and the table entries are justified by direct evaluations such as f7(alpha)=sqrt(7)u(2233sqrt(7))^2 and f12(alpha)=-sqrt(-3)(3-sqrt(-3))^2. The paper even corrects a typo in [2, Theorem 1.4] and a typo in [9], showing its inputs are used critically rather than as tautological premises. The classifications in [21] and [22] are published theorems about all rational elliptic curves, not results that already contain Table 2; they limit the candidate groups but do not determine the quadratic-field descriptions. Thus the central derivation is self-contained conditional on standard external classifications, and the residual risk is inherited completeness rather than circular reasoning.

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

No free parameters are fitted and no new entities are postulated. The central claim rests on published classification theorems and standard lemmas about twists and division polynomials. The only paper-specific computational ingredient is the Magma verification named in Remark 6.

assumptions (6)
  • domain assumption Olson's classification of rational CM torsion: Phi_CM(1) = {C1, C2, C3, C4, C6, C2 x C2}.
    Used in Section 3 to reduce the determination of E(Q)tors to roots of 2-, 3- and 4-division polynomials.
  • domain assumption Bourdon-Clark-Stankewicz [2, Theorem 1.4] classification of torsion subgroups of CM elliptic curves over quadratic fields.
    Used in Remark 5 and Section 4.2 to fix the finite list of possible growth groups; the paper corrects a typo in the source.
  • domain assumption Clark-Corn-Rice-Stankewicz [6] determination of Phi_CM(d) for 2 <= d <= 13.
    Supplies Phi_CM(2), the starting list for Theorem 1.
  • domain assumption Gonzalez-Jimenez-Tornero [21, Theorem 2] classification of torsion growth over quadratic fields for rational elliptic curves.
    Used in Section 4.2 to obtain Phi_Q(2,G), which is then intersected with CM restrictions.
  • standard math The identity E(Q(sqrt(d)))[n] is isomorphic to E(Q)[n] plus E^d(Q)[n] for odd n, where E^d is the quadratic twist.
    Stated in Section 2.3 (2); used to exclude C7 and C10 and to control odd-order growth.
  • ad hoc to paper The asserted factorizations of division polynomials over Q and quadratic fields, partly verified with Magma.
    Section 4.2 lists several finite algebraic checks without displaying full derivations; Remark 6 refers to Magma scripts on the author's webpage.

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Pith. "Pith review of Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields." pith.science (2026). https://pith.science/paper/RDWWNFZ5

@misc{pith2026190900637,
  author       = {Pith},
  title        = {Pith review of: Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDWWNFZ5}},
  note         = {Machine review of arXiv:1909.00637}
}
read the original abstract

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

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Works this paper leans on

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