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Explicit mock Heegner points prove that 2p and 2p² are rational cube sums for primes p ≡ 4 or 7 mod 9 when 2 is not a cube mod p, and settle BSD for the matching Mordell curves up to a 2-adic unit.

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Under 2 not a cube mod p, explicit mock Heegner points prove rank-1 BSD up to a 2-unit for E_{2q} and full BSD for the companion rank-0 curves E_{2q^{2}}.

T0 review reviewed 2026-07-30 challenge →

load-bearing objection Solid explicit mock-Heegner construction that settles rank-1 and nearly full BSD for the remaining 2p/2p^{2} Mordell curves in the 4,7 mod 9 classes under a clean local hypothesis. the 2 major comments →

arxiv 2607.26774 v1 pith:DDJBIP4T submitted 2026-07-29 math.NT

Explicit mock Heegner points and BSD formula on certain Mordell curves

Sumita Gunri , Somnath Jha , Dipramit Majumdar This is my paper
classification math.NT MSC 11G0511F1111D2511G40
keywords Mordell curvesmock Heegner pointsGross-Zagier formulaBSD conjecturerational cube sumscubic twistsring class fields
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

An integer is a rational cube sum when it equals a³ + b³ for rational a and b. That question is equivalent to whether the Mordell curve Y² = X³ − 432 n² has positive rank. For n = 2p or 2p² with p a prime congruent to 4 or 7 mod 9, the paper builds an explicit mock Heegner point on a related CM curve and proves the point is nontorsion precisely when 2 is not a cube modulo p. The resulting rational point shows that the rank is exactly 1, so 2q is a sum of two rational cubes. An explicit Gross–Zagier formula then matches the height of this point to the derivative of the L-function, confirming the rank part of BSD and showing that the full BSD formula holds up to a unit in Z[1/2]. For the companion rank-zero twists the same formula plus an external result gives the complete BSD prediction.

Core claim

For primes p ≡ 4 or 7 mod 9 with 2 not a cube in F_p, write q = p or p² accordingly. The Mordell curve E_{2q} : y² = x³ − 27 q² has analytic and algebraic rank exactly 1, so 2q is a rational cube sum; its Tate–Shafarevich group is finite and its order equals the BSD quotient up to a 2-adic unit. The companion rank-zero curve E_{2q²} satisfies the full BSD formula.

What carries the argument

An explicit mock Heegner point P₁ on the CM curve E : y² = x³ + 1, defined over the ring class field of conductor 6p, whose trace S₁ to K(∛p) is shown nontorsion by reduction modulo p against a fixed torsion point P′ = f′(ζ/6); cubic twisting then produces a nontorsion point on E_{2q}, and a variation of the Gross–Zagier formula relates its height to L′(1, E, χ).

Load-bearing premise

The proof that the constructed point is nontorsion relies on identifying a modular image of ζ/6 with an explicit torsion point by numerical approximation; if that identification is wrong the reduction argument fails.

What would settle it

For a concrete prime p ≡ 4 or 7 mod 9 with 2 not a cube mod p, compute the reduction of the constructed Heegner point modulo a prime above p and check whether it equals a 3-torsion point; alternatively evaluate L′(E_{2q}, 1) and the canonical height of the constructed rational point and test whether their ratio matches the predicted BSD quotient up to a power of 2.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Every prime p ≡ 4 or 7 mod 9 for which 2 is not a cubic residue yields an explicit rational point of infinite order on E_{2q}, so 2q = a³ + b³ for computable rationals a, b.
  • The rank part of BSD holds unconditionally for these E_{2q}, and |Sha| is determined up to a power of 2.
  • The companion curves E_{2q²} of root number +1 satisfy the complete BSD formula whenever the same cubic-residue hypothesis holds.
  • The same mock-Heegner construction supplies an explicit generator whose index in the Mordell–Weil group is a 3-adic unit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The numerical identification of f′(ζ/6) could be replaced by an algebraic proof that the eta-quotient evaluates to the listed torsion point, removing the only computational step.
  • When 2 is a cube mod p the same construction may still produce a point of infinite order on a quadratic twist, suggesting a route to the remaining cube-sum cases.
  • The method extends in principle to other conductors of the form 2p^k with root number −1, provided a suitable fixed torsion point and reduction congruence can be found.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper constructs an explicit mock Heegner point P1 on E: y^{2} = x^{3} + 1 over the ring class field H_{6p}, for primes p ≡ 4 or 7 mod 9 with 2 not a cube in F_p. After tracing to R1 (and S1 = 3R1) in E(L) with L = K(∛p), it proves nontorsion by reduction modulo primes above p, comparing the reduction of P1 to the Frobenius image of an auxiliary torsion point P' independent of p. Cubic twisting then yields a nontorsion point on E_{2q}(Q) (q = p or p^{2} according as p ≡ 4 or 7 mod 9), so rank E_{2q}(Q) = 1 and 2q is a rational cube sum. An explicit Gross–Zagier formula (via the CST1 variation) identifies L'(1, E_{2q}) with a multiple of the Néron–Tate height of the Heegner point; Kolyvagin–Gross–Zagier–Rubin give finiteness of Sha, and an index computation rules out 3-divisibility of the Heegner point in the Mordell–Weil lattice. The BSD formula for the rank-one curve is thereby verified up to a unit in Z[1/2]ˣ. For the companion rank-zero curves E_{2q^{2}} the same identity plus Burungale–Flach yields the full BSD formula.

Significance. The work settles the rank part of BSD (and the cube-sum problem) for an infinite family of Mordell curves E_{2p} and E_{2p^{2}} in the congruence classes p ≡ 4, 7 mod 9 under a mild local condition on 2, and obtains the full BSD formula for the associated rank-zero twists. The construction is fully explicit, the Galois action is tracked via Shimura reciprocity, and the nontorsion argument (reduction against a p-independent torsion point) is a genuine technical contribution relative to earlier mock-Heegner treatments of cube sums. The height formula is taken from a published variation of Gross–Zagier and specialized carefully; the 3-adic index computation is internal and does not assume BSD. These are concrete, checkable advances on a classical Diophantine problem and on BSD for CM curves of small conductor.

major comments (2)
  1. [Lemma 2.1 and Theorem 2.2] Lemma 2.1 identifies f'(ζ/6) with the explicit torsion point P' = (-u ζ^{2} ∛4, u(1+∛2)√-3) solely by SageMath numerical approximation, after noting that E(H6) ≅ (Z/6Z)^{2} has 36 torsion points. Every subsequent reduction identity in the proof of Theorem 2.2 (the explicit form of Q̄, the congruence R1 ≡ ±2Q̄ = (O,(0,±1)), and the contradiction ruling out R1 ∈ E(L)[3]) inherits this identification. While the candidate set is finite and eta quotients have integral q-expansions (Ligozat), the manuscript should either supply a rigorous algebraic identification of the CM point or document a reproducible high-precision verification that separates P' from the other 35 torsion points to a stated working precision. As written, the novel nontorsion step rests on an unreproducible floating-point match.
  2. [Theorem 2.2, display (2.2)] In the reduction computation leading to (2.2), the authors assert that 2·((-v^{2},0),(2v,-3)) equals (O,(0,±1)) in E(F_p)×E(F_p). The first component is immediate (2-torsion), but the second component 2·(2v,-3) = (0,±1) is not expanded. A short verification of the duplication formula (or a reference to the 2-division polynomial evaluated at x = 2v) would make the key congruence fully self-contained.
minor comments (6)
  1. [Title / running heads] The running header and title page contain spacing artefacts (“CER T AIN”, “CUR VES”, “f or”, “modulop”). These should be cleaned for the published version.
  2. [Table 1] Table 1 lists images of cusps under f'; a brief sentence confirming that the numerical approximations were cross-checked against the known torsion subgroup E(K)_tors would reassure the reader that the table is exact rather than approximate.
  3. [Proposition 1.1] In Proposition 1.1 the proof that E(L)[3^∞] = E(K)[3] uses irreducibility of φ3(x) = x^9 - 96x^6 + 48x^3 + 64 over K. A reference or one-line argument for that irreducibility would be helpful.
  4. [Section 3] The notation switches between E_{2q}, E^{(q)}_1 and E^{(q)}_2; a single sentence early in §3 recalling the precise Weierstrass models and the isogeny relating E_{2q} to E^{(q)}_1 would improve readability.
  5. [Proof of Theorem 0.1] Page 10, line after (3.8): “dim_{F3} Sel3(E^{(q)}_1/Q) = 2 and dim_{F3} Sel3(E^{(q)}_2/Q) = 1 [JMS2]” — the citation is appropriate, but stating the precise theorem number inside [JMS2] would aid the reader.
  6. [References] Several references (e.g., [ABS-BS], [LLT]) are still arXiv preprints; update to published versions where available.

Circularity Check

1 steps flagged

No structural circularity: Heegner nontorsion, Gross–Zagier specialization, and BSD index arguments are independent; only a minor non-load-bearing self-citation supplies 3-Selmer dimensions.

specific steps
  1. self citation load bearing [Proof of Theorem 0.1 (near eq. 3.9)]
    "As 2 is not a cube modulo p, dimF3 Sel3(E(q)1/Q)=2 and dimF3 Sel3(E(q)2/Q)=1 [JMS2] and we have, |X(E(q)1/Q)[3∞]|=|X(E(q)2/Q)[3∞]|=1."

    The 3-primary part of Sha is taken from the authors' own prior 3-Selmer papers rather than re-derived here. This is only a weak, non-central instance: the citation is used solely to match 3-adic valuations in the BSD quotient after the Heegner index has already been shown independently to be prime-to-3 (Lemma 2.4). It does not force rank, nontorsion, or the main L'/height identity.

full rationale

The derivation chain is self-contained in the usual number-theoretic sense. The mock Heegner point P1 is constructed from the CM point on X0(36) with explicit Galois action (Cor. 1.3, Prop. 1.5); nontorsion of R1/S1 is proved by reduction mod p against an explicit torsion candidate (Thm. 2.2), without assuming BSD or L-derivatives. The height–L' identity is obtained by specializing the published Cai–Shu–Tian variation of Gross–Zagier (Thm. 3.1, citing CST1), not by fitting or self-definition. Rank=1 and finiteness of Sha then follow from GZ+Kolyvagin+Rubin. The only author-overlapping citations used quantitatively are [MS2,JMS2] for dim Sel3, which pin |Sha[3∞]|=1 when controlling the 3-part of the BSD quotient; that input is an independent 3-descent computation and is not needed for the rank or nontorsion claims, nor does it force the Heegner index (Lemmas 2.3–2.4 already show 3∤a by a separate reduction contradiction). Rank-zero full BSD is imported from Burungale–Flach, an external theorem. No equation equals its input by construction, no parameter is fitted and re-predicted, and no uniqueness ansatz is smuggled in. Score 1 only for the minor, non-central self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 1 invented entities

The paper is pure arithmetic geometry. It imports class-field theory, Shimura reciprocity, the modular interpretation of X_0(36)≅E, the CST1 Gross–Zagier variant, Kolyvagin–Gross–Zagier–Rubin Euler-system machinery, Burungale–Flach for CM rank-0 BSD, and the authors’ earlier 3-Selmer computations. No continuous parameters are fitted. The only paper-specific analytic input is a numerical identification of one CM value of an eta quotient with an explicit torsion point.

axioms (8)
  • domain assumption Variation of the Gross–Zagier formula [CST1, Thm. 1.6] relating L'(1,E,χ) to the Néron–Tate height of the Heegner cycle on the CM curve E
    Invoked as a black box in §3 to obtain Theorem 3.1; the paper only specializes constants (periods, Petersson norm) via SageMath and twisting.
  • domain assumption Burungale–Flach theorem: full BSD for the relevant rank-0 CM Mordell curves once L(1)≠0
    Used in the proof of Theorem 0.2 to upgrade nonvanishing of L(1,E_{2q^{2}}) to the full BSD formula.
  • domain assumption Gross–Zagier, Kolyvagin, and Rubin: Heegner nontorsion ⇒ analytic rank 1, algebraic rank 1, and Sha finite
    Standard Euler-system input for the rank part of Theorem 0.1.
  • ad hoc to paper 2 is not a cube in F_p
    Standing hypothesis of Theorems 0.1–0.2 and of the nontorsion argument (Theorem 2.2); ensures inertness of primes above p in H_{6p}/H_{3p} and rules out R_1 torsion.
  • domain assumption dim_{F_3} Sel_3(E_{2q}/Q)=1 and companion Selmer dimensions from [MS2,JMS2]
    Used to bound rank ≤1 a priori and to kill the 3-primary part of Sha in the index comparison (3.9).
  • standard math Class field theory / Shimura reciprocity identifying Gal(H_{6p}/K) actions on the CM point P_0 with the stated matrices and Hilbert symbols
    Propositions 1.2–1.5; standard but load-bearing for the Galois eigenvalues of R_1 and S_1.
  • ad hoc to paper Numerical identification f'(ζ/6)=P'=(-u ζ^{2} ∛4, u(1+∛2)√-3) via SageMath approximation, using that E(H_6) is finite of known order
    Lemma 2.1; the only non-algebraic step in the nontorsion proof.
  • domain assumption ℓ-adic BSD results of Perrin-Riou, Kobayashi, and Li–Liu–Tian for good and ordinary primes, plus isogeny invariance of BSD
    Used to reduce the BSD formula for the rank-1 curve to a 3-adic index computation and to transfer from E^{(q)}_1 to E_{2q}.
invented entities (1)
  • Explicit mock Heegner point P_1 = f'(pζ/6) and its traces R_1, S_1 on E(L) independent evidence
    purpose: Provide a concrete nontorsion point whose height computes L'(1,E,χ) and which twists to a generator of E_{2q}(Q) up to finite index
    Constructed from standard CM theory on X_0(36); not a new physical or geometric object, but a new explicit arithmetic point for this family.

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Pith. "Pith review of Explicit mock Heegner points and BSD formula on certain Mordell curves." pith.science (2026). https://pith.science/paper/DDJBIP4T

@misc{pith2026260726774,
  author       = {Pith},
  title        = {Pith review of: Explicit mock Heegner points and BSD formula on certain Mordell curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DDJBIP4T}},
  note         = {Machine review of arXiv:2607.26774}
}
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abstract

For a natural number $a$, let $E_{2a}$ be the Mordell elliptic curve $X^3 +Y^3=2a$. We give an explicit construction of (mock) Heegner point on the Mordell curve $E_{2p}$ for a prime $p\equiv 4 \mod 9$ and $E_{2p^2}$ for a prime $p\equiv 7 \mod 9$, under the assumption that $2$ is not a cube modulo $p$. We also verify the explicit Gross-Zagier formula for these curves and go on to show that the BSD formula holds for these curves up to a $2$-adic unit. Using a result of Burungale-Flach, we show that the full BSD formula holds for the rank zero curve $E_{2p}$ for $p\equiv 7 \mod 9$ and $E_{2p^2}$ for $p\equiv 4 \mod 9$, whenever $2$ is not a cube modulo $p$.

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This paper was first reviewed by grok-4.5 on July 30, 2026.