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REVIEW 3 major objections 4 minor 25 references

Mazur's growth number conjecture and congruences

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Pith's one-line read Mazur's rank-growth conjecture survives mod p congruences

desk verdict A genuinely new congruence-propagation result with a real theorem, but Proposition 3.7 has a load-bearing gap in the unit-leading-coefficient step and a missing hypothesis in the final invocation. read the letter →

arxiv 2505.19542 v1 pith:MPIVQHC6 submitted 2025-05-26 math.NT

Anwesh Ray This is my paper · ORCID
classification math.NT MSC 11R2311G05
keywords MazurgrowthnumberconjectureSelmergroupsIwasawatheoryellipticcurvesmodpcongruencesHidafamiliesMordell-Weilranksimaginaryquadraticfields
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

The paper proves a conditional propagation theorem: once Mazur's growth number conjecture is available for a base elliptic curve E, the same boundedness of Selmer coranks (and hence of Mordell-Weil ranks) holds for any modular form f whose mod p Galois representation is isomorphic to E[p]. Under explicit hypotheses—including the two-variable Iwasawa main conjecture, a rank-one Selmer condition, and local tameness—the Selmer group of E over every non-anticyclotomic Z_p-extension is cotorsion with μ=0 and λ=1, and this property transfers to f by a Greenberg–Vatsal style comparison of residual Selmer groups. The upshot is that elliptic curves E' that are p-congruent to E have bounded ranks in such towers, and the same applies to all weight-k specializations of a Hida family that contain the base form.

What carries the argument

The argument is carried by the two-variable Iwasawa main conjecture (IMC), the assertion that the characteristic ideal of the dual Selmer group over the $Z_p^{2}$-extension equals the ideal generated by the two-variable p-adic L-function L_p(X,Y). The proof combines this with the Perrin-Riou/Howard–Castella–Disegni identity relating the derivative of L_p to the Λ-adic height pairing of Heegner points, and with a numerical criterion that identifies the tameness condition (tEC) with μ=0, λ=1 in the cyclotomic direction. Specializing along any non-anticyclotomic direction (a,b)≠(0,1), the leading coefficient of L_{a,b} is a p-adic unit, forcing the characteristic ideal to be (T), i.e., μ=0 and λ=1; a Greenberg–Vatsal-type comparison of residual Selmer groups then transfers this cotorsion property from E to any p-congruent modular form f.

What would settle it

Compute the Perrin-Riou Λ-adic height pairing ⟨z_∞, z_∞⟩_0 for a rank-one elliptic curve satisfying (HH) and (tEC); the proof requires this pairing to be a p-adic unit. Finding a curve where the pairing vanishes while the other hypotheses hold would break the step that every non-anticyclotomic specialization has λ=1, and with it the propagation to congruent forms.

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Extended reading notes

Core claim

The central claim is Theorem 3.8: for an elliptic curve E/Q with good ordinary reduction at p≥5 and an imaginary quadratic field K, if (E,K,p) satisfies the Heegner hypothesis (HH), the two-variable Iwasawa main conjecture (IMC), and the tameness criterion (tEC), and if f is a p-ordinary eigencuspform with A_f[ϖ] ≅ E[p]⊗κ, then for every non-anticyclotomic Z_p-extension K_{a,b} in which the primes above p are totally ramified, the Greenberg Selmer group $Sel^{{Gr}}$(A_f/K_{a,b}) has finite Z_p-corank. In the weight-2 case where f gives an elliptic curve E', the Mordell-Weil rank of E'(K_{a,b}) is finite and in fact bounded independently of the layer. The paper also establishes Proposition 3.7, which is Mazur's conjecture for the base curve: for every such specialization, the Selmer group is cotorsion with μ=0 and λ=1, giving corank 1 in each finite layer, and hence the growth-number prediction c=0 for all non-anticyclotomic towers.

Load-bearing premise

The proof assumes the two-variable Iwasawa main conjecture for the base elliptic curve E, an unproved equality between a p-adic L-function and a characteristic ideal; if that equality fails for E, the derivation of μ=0, λ=1 for all specializations collapses.

Editorial extensions

If this is right

  • Mazur's growth number conjecture holds for (E,K,p) whenever (HH), (IMC), and (tEC) are satisfied, with growth constant c=0 for every non-anticyclotomic Z_p-extension.
  • For any elliptic curve E' with E'[p]≅E[p], the Mordell-Weil rank of E'(K_{a,b}) is bounded as n grows, for every non-anticyclotomic tower in which p and \bar p are totally ramified.
  • All classical weight-k specializations (k≡2 mod p−1) of a Hida family containing f_E have finite Greenberg Selmer corank over the same towers.
  • The explicit example E=37a1, E'=1406g1 at p=5 provides a proved instance where the rank boundedness holds.

Reading between the lines

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

  • If the two-variable Iwasawa main conjecture is proved in the required cases, the conditional result becomes unconditional for all such triples, making the growth-number conjecture a consequence of the main conjecture rather than a separate phenomenon.
  • The same strategy is silent about the anticyclotomic direction (a,b)=(0,1), where the growth constant can be 1 or 2; one might test whether congruences also propagate the larger growth numbers there, but the paper's method deliberately avoids that case.
  • The key numerical invariant ⟨z_∞,z_∞⟩_0 is computable in principle, as the paper itself notes; verifying nonvanishing for more examples would give computational certificates for Mazur's conjecture in those cases.
  • Since the hypotheses are explicit, one could systematically search for further p-congruent curve pairs (like 37a1 and 1406g1) to enlarge the database of cases where bounded ranks are known unconditionally.
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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

3 major / 4 minor

Summary. The paper investigates Mazur's growth number conjecture for Z_p-extensions of an imaginary quadratic field, under explicit Iwasawa-theoretic hypotheses, and then propagates the result to modular forms congruent to a fixed elliptic curve mod p. The main results are Proposition 3.7, asserting that under (HH), (IMC), and (tEC) every non-anticyclotomic Z_p-extension has μ=0, λ=1 Selmer group and that Conjecture 1.1 holds, and Theorem 3.8, asserting finite corank of Greenberg Selmer groups for congruent modular forms, with consequences for Mordell-Weil ranks. The proof uses the 2-variable Iwasawa main conjecture, the p-adic Gross-Zagier formula, and a congruence argument of Greenberg-Vatsal type.

Significance. If the technical gaps were repaired, the paper would establish a useful conditional result: Mazur's conjecture for rank-1 elliptic curves under standard hypotheses, and its propagation under p-congruences to Hida-style families and congruent elliptic curves. The paper is honest in stating its conditional hypotheses (IMC, tEC, HH) and does not fit parameters; the congruence-propagation mechanism (Propositions 2.6 and 2.7) is a legitimate contribution. However, the central derivation of the μ=0, λ=1 statement for all non-anticyclotomic lines is incomplete, and the final application of Kundu-Lei's theorem omits required hypotheses. The significance is therefore conditional on a substantial repair.

major comments (3)
  1. [§3.2, Proposition 3.7(1)] The proof that α_{a,b} ∼ ⟨z∞,z∞⟩0 for every (a,b)≠(0,1) is not justified. For a line with direction (a,b), the linear coefficient of L_{a,b}(T) is the directional derivative of Lp(X,Y) at (0,0) in the direction (a,b), namely a·(∂Lp/∂X)(0,0) + b·(∂Lp/∂Y)(0,0) up to normalization. Theorem 3.1 identifies only the Y-derivative at Y=0 (the anticyclotomic direction) with the Heegner height pairing; it gives no information about (∂Lp/∂X)(0,0) or about values on other lines. The argument proves c2 := (∂Lp/∂Y)(0,0) is a p-adic unit, but unless c1 := (∂Lp/∂X)(0,0) = 0, the linear form c1 a + c2 b vanishes on a projective line different from (0,1), and for that line the specialization has leading coefficient divisible by p or vanishing, so the conclusion μ=0, λ=1 cannot follow. The paper excludes only the single point (0,1) and supplies no arithmetic reason that c1 = 0 or that the zero direction of the linear part is exactly (0,1). Since Theorem 3.8 invokes Proposition 3.7 for every (a,b)≠(0,1), this is a load-bearing gap.
  2. [§3.2, Proposition 3.7(2)] The inference 'Since ⟨z∞,z∞⟩∞ ≠ 0, it follows from Theorem 3.3 that Conjecture 1.1 holds' omits two hypotheses of Theorem 3.3: the elliptic curve E must have analytic rank 1 and must not have complex multiplication. The assumptions of Proposition 3.7 are (HH), (IMC), (tEC), and E(K)[p]=0; among these, (tEC) includes corank Zp Selp∞(E/K) = 1, but no analytic-rank condition or no-CM condition is stated or derived. As stated, Theorem 3.3 cannot be applied, so part (2) of Proposition 3.7 is not established.
  3. [§3.1, Theorem 3.1 / §3.2, Proposition 3.7] Even for the cyclotomic line (1,0), the deduction that ⟨z∞,z∞⟩0 ∈ Z_p^× from α ∈ Z_p^× requires more justification. Theorem 3.1 identifies ∂Lp/∂Y(X,0) with the height pairing only as principal ideals in Qp⊗Λac, so the constant terms agree only up to a p-adic unit of unknown valuation. Since the proof of Proposition 3.7 uses this to conclude the pairing value is a p-adic unit, an integrality or unit argument is needed that is not provided.
minor comments (4)
  1. [§3.2, proof of Proposition 3.7] The phrase 'Since L1,0 divides (T)' is inaccurate; from (IMC) and the equality of ideals one gets that L1,0 generates (T), i.e., L1,0 = u(T)·T for a unit u(T) in Λcyc. Please correct the wording.
  2. [§3.1, Example 3.9] The set of vexing primes V is defined as all primes ℓ ≡ -1 mod p satisfying the residual conditions, which in principle could be infinite; the verification that V is empty for E=37a1 because 37 ≢ -1 mod 5 appears to assume that only primes dividing the conductor are relevant. Please clarify the scope of V in Theorem 3.2.
  3. [§3.2, Theorem 3.8] The application of Proposition 2.7 requires H0(K, Af[̟]) = 0; although this likely follows from E(K)[p] = 0 together with Af[̟] ≅ E[p]⊗κ, the implication should be stated explicitly.
  4. [§3.3, Hida families] The condition 'for each integer k > 0 with k ≡ 2 (mod p−1)' should read 'k ≥ 2' since weight-k cuspidal eigenforms require k ≥ 2.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main derivation is a genuine conditional implication from (IMC)+(tEC), with only minor self-citation and some unproved analytic steps, not circular reductions.

full rationale

The paper's central chain is conditional: Theorem 3.8 assumes (HH), (IMC), and (tEC). Given (IMC), the characteristic ideal of the dual Selmer group over K_infty equals the two-variable p-adic L-function, so each specialization line satisfies I_{a,b}=(L_{a,b}) via Proposition 3.6. Theorem 2.8, cited to the author's prior [MR24], converts (tEC) into mu=0, lambda=1 on the cyclotomic line, giving I_{1,0}=(T); Theorem 3.1 then identifies the linear coefficient with the Heegner height, so the nonvanishing of the height is derived rather than assumed. The transfer to f via Proposition 2.7 is the standard Greenberg--Vatsal congruence argument. Reliance on the unproved (IMC) is reliance on a conjecture, not circularity. There is one self-citation for Theorem 2.8, but the cited result is a standard Perrin--Riou/Schneider numerical criterion and does not presuppose the paper's conclusions; under the stated rules, an independent cited criterion does not raise the circularity score. The genuine weaknesses are correctness gaps, not circularity: Proposition 3.7 extends the leading-coefficient relation alpha_{a,b} ~ <z_infty,z_infty>_0 to every non-anticyclotomic line by 'same arguments in loc. cit.' without supplying the argument (Theorem 3.1 controls only one line), and Theorem 3.3 is invoked without explicitly listing the analytic-rank-1 and no-CM hypotheses. These are missing proofs and missing hypotheses, not reductions of the output to the input, so they should be weighed as correctness risk rather than circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the unproved Iwasawa main conjecture and on a list of explicit numerical conditions, rather than on fitted parameters or newly invented objects.

assumptions (4)
  • domain assumption The two-variable Iwasawa main conjecture (IMC) for (E,K,p)
    Invoked in Proposition 3.7 to identify the characteristic ideal of the dual Selmer group with the p-adic L-function. This is a deep unproved conjecture and the main load-bearing assumption.
  • domain assumption The numerical criterion (tEC), including corank Selp∞(E/K)=1, p∤Reg_p(E/K), X(E/K)[p∞]=0, and vanishing of local contributions
    These are explicit verifiable conditions, but some (e.g., X(E/K)[p∞]=0 and p∤Reg_p) are conjectural in general and not known for arbitrary E. Used via Theorem 2.8 to get μ=0, λ=1.
  • domain assumption The Heegner hypothesis (HH): all primes dividing N p split in K
    Ensures existence of Heegner points and finite decomposition of relevant primes; used throughout Sections 2 and 3.
  • domain assumption Residual isomorphism Af[̟] ≅ E[p]⊗κ in Theorem 3.8
    The congruence hypothesis that connects f to E; it is an explicit condition to be verified, not a general fact.

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Pith. "Pith review of Mazur's growth number conjecture and congruences." pith.science (2026). https://pith.science/paper/MPIVQHC6

@misc{pith2026250519542,
  author       = {Pith},
  title        = {Pith review of: Mazur's growth number conjecture and congruences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPIVQHC6}},
  note         = {Machine review of arXiv:2505.19542}
}
abstract

Motivated by the work of Greenberg-Vatsal and Emerton-Pollack-Weston, I investigate the extent to which Mazur's conjecture on the growth of Selmer ranks in $\mathbb{Z}_p$-extensions of an imaginary quadratic field persists under $p$-congruences between Galois representations. As a first step, I establish Mazur's conjecture for certain triples $(E, K, p)$ under explicit hypotheses. Building on this, I prove analogous results for Greenberg Selmer groups attached to modular forms that are congruent mod $p$ to $E$, including all specializations arising from Hida families of fixed tame level. In particular, I show that the Mordell-Weil ranks in non-anticyclotomic $\mathbb{Z}_p$-extensions of $K$ remain bounded for elliptic curves $E'$ such that $E[p]$ and $E'[p]$ are isomorphic as Galois modules.

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