{"id":"2a541065-3f46-4277-a4e9-15a1585cf512","arxiv_id":"2505.19542","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under the Iwasawa main conjecture and explicit numerical conditions, Mazur's growth number conjecture holds for non-anticyclotomic Z_p-extensions and propagates to p-congruent modular forms.","lead":"The paper proves conditional results about how the ranks of elliptic curves grow in certain towers of number fields, and shows these results transfer to modular forms that are congruent to the curve modulo p. The conclusions depend on the unproved Iwasawa main conjecture and on explicit numerical conditions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Prop. 3.7(1) rests on the assertion that L_{a,b} has p-adic-unit leading coefficient for every (a,b)≠(0,1).","rationale":"The reader identified (IMC) as the weakest assumption, but (IMC) is an explicit hypothesis, so a conditional theorem may legitimately assume it. The reader also noted, in passing, that the interpolation step for the leading coefficient of L_{a,b} is only sketched; that is the point I find most load-bearing. The proof needs the leading coefficient of L_{a,b} to be a p-adic unit for every non-anticyclotomic line. For a two-variable power series with nonlinear terms, what is actually determined by knowing L_{1,0}=(T) is the behavior of the linear part along one line. The remaining directions are controlled by the partial derivative in the excluded direction; nothing in the cited theorem forces that derivative to be compatible with every specialization in the way asserted. Over Qp, a nonzero linear form has exactly one zero direction, and near that direction in the p-adic topology the coefficients are divisible by higher powers of p. The paper gives no argument that the zero direction is exactly the excluded anticyclotomic line, nor that the lines near it are not permitted. This is not a dispute with the Iwasawa main conjecture; it is an internal gap in the proof of Prop. 3.7(1), and Theorem 3.8 inherits it. A conditional verdict is appropriate: if the author can prove the needed leading-coefficient identity for all allowed lines, or restrict the theorem to the lines where it holds, the main congruence-propagation idea may survive. I do not endorse REJECT because the underlying Greenberg–Vatsal-style propagation (Prop. 2.7) appears sound and the issue may be repairable by adding the missing interpolation argument or a hypothesis excluding the bad residue class. My disagreement with the reader is partial because the reader mentioned the interpolation step but did not make it the primary concern.","tokens_in":13619,"tokens_out":26621,"duration_ms":267867,"concrete_test":"Use the explicit setting of Example 3.9 (E=37a1, p=5, a suitable imaginary quadratic K for which Burungale–Castella–Skinner gives (IMC) and the Heegner hypothesis holds) and compute the two-variable p-adic L-function or its characteristic-ideal counterpart F(X,Y). Specialize F along the lines (a,b)=(p^n,1) for n≥1, using the quotient relation (1+X)^{p^n}(1+Y)=1 and T=X. If the leading coefficient of F_{p^n,1}(T) is not a p-adic unit for some n, then Prop. 3.7(1) and Theorem 3.8 fail as stated. If such a computation is not readily available, the same check on the generic series F(X,Y)=X+Y is decisive against the proof's stated interpolation step: its specialization along (p^n,1) has leading coefficient 1−p^n, divisible by p^n, showing that the claimed α_{a,b}∼⟨z∞,z∞⟩0 cannot hold for a generic nonzero linear part.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Prop. 3.7(1) is the engine of Theorem 3.8: it claims Selp∞(E/Ka,b) is cotorsion with μa,b=0 and λa,b=1 for every line (a,b)≠(0,1). The proof uses IMC and Theorem 2.8 to get I1,0=(T), hence L1,0(T)=uT, and cites Theorem 3.1 to infer that the coefficient αa,b of the linear term of L_{a,b}(T) is αa,b∼⟨z∞,z∞⟩0 for all (a,b). This inference is not justified. If F(X,Y)=Lp(X,Y) has nonzero linear part — as the proof itself needs, via Theorem 3.1 — then the leading coefficient of the specialization F_{a,b}(T) is a p-adic analytic function on P1(Zp). A nonzero linear form on Zp^2 vanishes on exactly one projective line, and on the entire residue disk of that line the specialized linear coefficient is divisible by p. Unless the vanishing line is (0,1) and its whole residue disk is excluded, infinitely many allowed lines have non-unit leading coefficient, giving μ>0 or λ>1. The paper excludes only the single point (0,1) and gives no arithmetic reason that the zero direction of the linear part is exactly that point or that nearby lines are disallowed. Since Theorem 3.8 invokes Prop. 3.7(1) directly for every such line, this is a load-bearing gap in the central argument, independent of the depth of (IMC).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13783,"tokens_out":16191,"duration_ms":140441,"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":[{"comment":"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.","section":"§3.2, Proposition 3.7(1)"},{"comment":"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.","section":"§3.2, Proposition 3.7(2)"},{"comment":"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.","section":"§3.1, Theorem 3.1 / §3.2, Proposition 3.7"}],"minor_comments":[{"comment":"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.","section":"§3.2, proof of Proposition 3.7"},{"comment":"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.","section":"§3.1, Example 3.9"},{"comment":"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.","section":"§3.2, Theorem 3.8"},{"comment":"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.","section":"§3.3, Hida families"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious but incomplete conditional contribution. The main technical gap—the behavior of the linear part of the p-adic L-function under all specializations—affects the validity of the central theorem. I recommend major revision rather than rejection because the gap is a specific missing argument that could potentially be repaired by adding an explicit hypothesis on the cyclotomic derivative or by proving it under the stated assumptions. The author should also carefully re-read the hypotheses of the Kundu-Lei theorem and the exact formulation of the p-adic Gross-Zagier formula used."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe paper's actual contribution is Theorem 3.8: assuming Mazur's growth number conjecture in the strong form of (IMC) and (tEC) for one pair (E,K,p), the conjecture propagates to every p-ordinary modular form whose residual representation is E[p], including Hida family specializations. That is a genuine step beyond Greenberg–Vatsal and Emerton–Pollack–Weston, and the congruence argument via residual Selmer groups (Prop 2.7) is natural and reads fine. The explicit 5-congruent example is a nice touch.\n\nThe problems sit in Proposition 3.7, which is the engine. Part (2) invokes Kundu–Lei's Theorem 3.3 to conclude Conjecture 1.1 for (E,K,p), but that theorem assumes analytic rank 1 and non-CM, and neither appears in the statement of Prop 3.7 or Theorem A. That is an objective mismatch between statement and proof.\n\nMore serious is part (1). The proof claims that for every (a,b)≠(0,1) the linear coefficient α_{a,b} of L_{a,b}(T) is a p-adic unit and satisfies α_{a,b}∼⟨z∞,z∞⟩_0. But Lp(X,Y) has linear part cX+dY; along the line (a,b) the leading term is (ca+db)T. You get a unit only if ca+db is a unit. That is a p-adic-analytic condition on P1(Zp). Unless the zero direction is exactly (0,1) and all lines in the residue disk of (0,1) are excluded, infinitely many allowed lines have leading coefficient divisible by p. The paper excludes only the single point (0,1), not the residue class, and gives no arithmetic reason the zero direction is what it needs to be. Theorem 3.8 leans directly on this step, so this is load-bearing.\n\nNone of this is fatal to the research program—the gaps are identifiable and likely fixable with stronger hypotheses—but the current statements overreach the proof.\n\nFor you: this is a paper for a specialist in Iwasawa theory who wants to see the congruence propagation idea developed. It deserves a serious referee because the main idea is valuable and the flaws are precise, not vague. I'd expect the referee to ask for major revision.","headline":"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.","tokens_in":14511,"tokens_out":6982,"would_cite":false,"duration_ms":67554,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R23","11G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Mazur's rank-growth conjecture survives mod p congruences","keywords":["Mazur growth number conjecture","Selmer groups","Iwasawa theory","elliptic curves","mod p congruences","Hida families","Mordell-Weil ranks","imaginary quadratic fields"],"falsifier":"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.","tokens_in":13225,"feed_emoji":"🔢","tokens_out":9541,"duration_ms":52165,"temperature":0.7,"pith_summary":"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.","feed_headline":"Mazur's rank-growth conjecture survives mod p congruences","feed_subtitle":"The proof transfers the base curve's p-adic L-function identity to all forms with the same mod p Galois representation.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the method of comparing Iwasawa invariants under mod p congruences via residual Selmer groups; the proof of Proposition 2.7 is modeled on it.","marker":"[GV00]"},{"why":"Provides the numerical criterion (Theorem 2.8) equating the tameness condition (tEC) with μ=0, λ=1 for the cyclotomic Selmer group.","marker":"[MR24]"},{"why":"Proves Mazur's growth number conjecture in the rank-one case (Theorem 3.3) and contains the identity used to identify the leading coefficient of L_{a,b} with the Perrin-Riou height pairing.","marker":"[KL25]"},{"why":"Proves the p-adic Gross-Zagier formula (Theorem 3.1) identifying the derivative of the two-variable L-function with the Λ-adic height pairing, used to show the leading coefficient is a unit.","marker":"[Dis17]"},{"why":"Establishes the two-variable Iwasawa main conjecture under explicit hypotheses (Theorem 3.2), making (IMC) available for the example and for conditional results.","marker":"[BCS25]"},{"why":"Supplies structural results on Selmer modules over the Z_p^2-extension (semisimplicity and restriction isomorphism, Propositions 3.5 and 3.6) used to identify characteristic ideals at specializations.","marker":"[GLHKL24]"}],"fun_headline_variants":["Mod p congruences preserve Mazur's rank growth","Selmer ranks of congruent modular forms stay bounded","Mazur's conjecture proven for Hida family specializations","Elliptic curves with mod p isomorphisms share rank bounds","Non-anticyclotomic Selmer ranks stay finite under congruence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Mod p congruences preserve Mazur's rank growth","Selmer ranks of congruent modular forms stay bounded","Mazur's conjecture proven for Hida family specializations","Elliptic curves with mod p isomorphisms share rank bounds","Non-anticyclotomic Selmer ranks stay finite under congruence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000803,"raw_usage":{"total_tokens":3537,"prompt_tokens":960,"completion_tokens":2577,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":2505}},"tokens_in":576,"tokens_out":2577,"duration_ms":14273,"temperature":1.0,"reasoning_tokens":2505,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:12:18.699593+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}