On Howard's main conjecture and the Heegner point Kolyvagin system

Murilo Zanarella

REVIEW 3 major objections 3 minor 22 references

On Howard's main conjecture and the Heegner point Kolyvagin system

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a primitive Heegner-point Kolyvagin system upgrades Howard's divisibility to the full Howard Main Conjecture, and it establishes an equivalence between that conjecture and primitivity plus a Tamagawa-factor condition.

desk verdict A serious, well-structured paper that upgrades Howard's divisibility to equality under a primitivity hypothesis; the one load-bearing spot is the terse mod-p congruence bridge, which needs a careful check. read the letter →

arxiv 1908.09197 v1 pith:HXUMOCKL submitted 2019-08-24 math.NT

Murilo Zanarella This is my paper · ORCID

classification math.NT MSC 11R2311G0511G40

keywords Howard'sMainConjectureHeegnerpointsKolyvaginsystemsIwasawatheoryellipticcurvesanticyclotomicextensionp-adicL-functionsprimitivity

topics Birch and Swinnerton-Dyer

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's central claim is that Howard's Main Conjecture for Heegner points—an equality of characteristic ideals in anticyclotomic Iwasawa theory—follows from a single explicit condition: the usual finite-level Kolyvagin system $\kappa$ is primitive, meaning its reduction modulo $p$ is nonzero. The proof works by refining Howard's Kolyvagin-system formalism with an error term $d(\kappa)$ that measures the gap between a Selmer module's length and the length of the quotient by the Heegner class; this gap vanishes exactly when $\kappa$ is primitive. The author then shows, through mod-$p$ congruences that require $p$ to be non-anomalous, that primitivity of $\kappa$ forces the Iwasawa-theoretic Heegner system $\kappa^{Hg}$ to be $\Lambda$-primitive, which is precisely what upgrades the known divisibility to equality. The paper also proves a partial converse: assuming Howard's Main Conjecture, $\kappa$ is primitive exactly when no split prime Tamagawa factor is $p$-divisible, and a full equivalence under Assumptions 1.1.7 and 1.1.1 states that $\kappa$ is primitive precisely when Howard's Main Conjecture and the Tamagawa condition both hold. If correct, the paper turns a conjecture about $p$-adic $L$-functions and Selmer groups into a statement about indivisibility of Heegner points, extending known cases to analytic rank greater than one and to non-semistable curves.

What carries the argument

The engine of the proof is a refinement of the Kolyvagin-system formalism for conjugate-self-dual Galois representations. For a Kolyvagin system $\kappa$ over a discrete valuation ring, the paper proves the precise error formula $\operatorname{length}(M)=\operatorname{length}(H^1_F(K,T)/R\kappa_1)-d(\kappa)$, where $M$ is the torsion module in the structure $H^1_F(K,A)\simeq D\oplus M\oplus M$, and $d(\kappa)$ is an integer attached to $\kappa$ with $d(\kappa)=0$ exactly when $\kappa$ is primitive. This is the object that converts the known inequality into an equality: the divisibility in Howard's Main Conjecture is the inequality side, and the error term vanishes precisely under primitivity. To get from finite-level primitivity to the Iwasawa-theoretic condition, the paper uses the mod-$p$ congruences of Propositions 3.2.2 and 3.2.3, which identify the reduction of the $\Lambda$-adic Heegner system $\kappa^{Hg}$ with a nonzero scalar multiple of the reduction of $\kappa$; in the split ordinary case the scalar is $(a_p-1)^2$ and in the supersingular case it is $-2$, and these are exactly where the non-anomalous hypothesis enters.

What would settle it

Take an elliptic curve $E/\mathbb{Q}$, a prime $p\ge 5$, and a quadratic imaginary field $K$ satisfying the theorem's hypotheses with $\kappa$ primitive, and compute the mod-$p$ reductions of $\kappa$ and of the Iwasawa-theoretic class $\kappa^{Hg}$ at a square-free level where the Selmer group has rank one. The central bridge predicts the exact congruences $D_0D_m\Phi P[m]\equiv(a_p-1)^2D_0D_mP[m]\pmod p$ in the split ordinary case and $D_0D_mP_0[m]\equiv-2D_0D_mP[m]\pmod p$ in the supersingular case; exhibiting any curve where one of these fails, or where $\kappa$ is primitive yet the two characteristic ideals have different $p$-adic valuations, would refute Theorem A.

Watch

Extended reading notes

Core claim

Concretely, under the paper's Assumptions 1.1.1 and 1.1.3, the non-anomalous condition, and primitivity of the Heegner Kolyvagin system $\kappa$, Theorem A establishes the full statement of Howard's Main Conjecture: the torsion $\Lambda$-module $M$ satisfies $\operatorname{char}(M)=\operatorname{char}(M)^\iota$, the Selmer group $X$ is pseudo-isomorphic to $\Lambda\oplus M\oplus M$, and $\operatorname{char}(M)=\operatorname{char}(H^1_{F_\Lambda}(K,T)/\Lambda\kappa^{Hg}_1)$. In other words, the divisibility obtained in the original and supersingular formulations is upgraded to equality. The paper also proves a converse direction: assuming Howard's Main Conjecture, $\kappa$ is primitive if and only if no split prime Tamagawa factor is $p$-divisible, and an equivalence theorem states that $\kappa$ is primitive exactly when Howard's Main Conjecture and that Tamagawa condition both hold. The route combines the refined Kolyvagin-system formalism with an anticyclotomic control theorem and an explicit reciprocity law for Heegner points, which ties the BDP $p$-adic $L$-function main conjecture to the index of the Heegner classes.

Load-bearing premise

The load-bearing premise is the exactness of the mod-$p$ congruences (Propositions 3.2.2 and 3.2.3) that identify the reduction of the Iwasawa-theoretic Heegner system $\kappa^{Hg}$ with a nonzero scalar multiple of the reduction of the finite-level system $\kappa$; the paper's derivation of these congruences is terse, and any $p$-factor error there would destroy the bridge from primitivity of $\kappa$ to $\Lambda$-primitivity of $\kappa^{Hg}$.

Editorial extensions

If this is right

Howard's Main Conjecture becomes a theorem in every case where primitivity of the Heegner Kolyvagin system is known, including curves of analytic rank greater than one and curves that are not semistable.
The equality of characteristic ideals $\operatorname{char}(M)=\operatorname{char}(H^1_{F_\Lambda}(K,T)/\Lambda\kappa^{Hg}_1)$ follows from the error-term formula once $\kappa^{Hg}$ is known to be $\Lambda$-primitive.
Theorem C gives an arithmetic criterion: under the stated hypotheses, Howard's Main Conjecture holds exactly when $\kappa$ is primitive and no split prime Tamagawa factor is $p$-divisible.
In the supersingular reduction case, the same argument proves the `+`-version of the main conjecture under the analogous hypotheses.
The proof shows that an explicit reciprocity law can be used to read primitivity of $\kappa$ from the $p$-adic valuation of a BDP $L$-value, connecting the main conjecture to classical Kolyvagin structure theorems.

Reading between the lines

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

If primitivity is the sharp hypothesis, the same strategy may prove Howard's Main Conjecture for abelian varieties of $\mathrm{GL}_2$ type or higher-weight modular forms wherever an explicit reciprocity law is available.
The error term $d(\kappa)$ may coincide with the classical Kolyvagin-system divisibility exponent in a wider range of Selmer structures; proving equality would make the formalism uniform and give a single primitivity criterion.
Because the proof bypasses semistability, a natural test is to compute both characteristic ideals and the error term on a non-semistable curve of analytic rank 2; a failure would localize the obstruction to the mod-$p$ congruences rather than to the underlying Iwasawa theory.
The equivalence suggests a purely geometric reformulation: the main conjecture may be equivalent to the indivisibility of Heegner points together with the absence of split Tamagawa factors, a statement approachable by congruences between modular forms.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new thing here is an error term d(kappa) in Howard's Kolyvagin-system formalism, which upgrades the usual divisibility to an equality exactly when the Heegner Kolyvagin system is primitive. That is a real improvement, and Theorem A follows cleanly from it. The formal core in Section 2 is coherent: Theorem 2.3.6 refines Howard's inequality in a natural way, and the adaptation of the Mazur–Rubin argument to get equality from Lambda-primitivity is credible. The paper is also honest—it flags Assumption 1.1.1 as likely removable and does not hide that Theorem B is conditional on Howard's main conjecture.

The weakest point is the bridge between the two primitivity notions. Propositions 3.2.2 and 3.2.3 assert the mod-p congruences that transport primitivity from the finite-level Kolyvagin system kappa to the Iwasawa-level system kappa^Hg, and the derivations are compressed into single lines with no appendix revisiting them. The stress-test note correctly points out a sign issue in the inert ordinary case: direct reduction of Phi at p gives (1 - a_p^2), not (a_p^2 - 1). That sign is a p-adic unit under (not anom), so it is harmless for primitivity, but it illustrates exactly how easy it is to drop a unit. A missing p-factor anywhere in those three congruences would break Theorem A, because the whole argument reduces to choosing m where the reduced class is nonzero. I do not see an actual error, but this is the spot that needs the most careful checking.

Two other soft spots, both minor in proportion. Theorem 4.2.1 leans on an internal step of [JSW17] that is not reproduced; that is a verification burden, not an error. Some local computations at primes above p in the control theorem are terse. None of these strike me as load-bearing in the way the congruences are.

Who this is for: Iwasawa theorists working on Heegner points, especially anyone using Howard's formalism. It deserves a serious referee. An expert should spend real time on Propositions 3.2.2/3.2.3 and on the JSW17 application; if those check out, this is accept-level work. I would send it to peer review, not desk reject.

Referee Report

3 major / 3 minor

Summary. The paper studies Howard's main conjecture for Heegner points in the anticyclotomic Iwasawa theory of an elliptic curve. Its central result, Theorem A, says that if the finite-level Heegner Kolyvagin system κ is primitive and the usual running hypotheses hold (including the non-anomalous condition), then Howard's Main Conjecture is true, i.e. the divisibility obtained by Howard is upgraded to equality and the Selmer module has the predicted pseudo-isomorphism type. Theorem B gives a conditional converse: assuming Howard's Main Conjecture and a split-ordinary or analytic-rank-one situation, κ is primitive exactly when the split Tamagawa factors are p-indivisible. Theorem C combines these into an equivalence between primitivity of κ, Howard's Main Conjecture, and the absence of split Tamagawa factors. The main new technical ingredients are an improvement of Howard's Kolyvagin-system formalism with an error term d(κ) satisfying length(M)=length(H^1_F(K,T)/Rκ_1)-d(κ), and a comparison between the finite-level and Iwasawa-theoretic Heegner Kolyvagin systems modulo p.

Significance. If the results are correct, the paper is substantial: it gives a new route from a primitivity statement to the full Howard/Perrin-Riou main conjecture, it covers cases of analytic rank greater than one, and it does not require semistability of the elliptic curve. The improved Kolyvagin-system formalism with the explicit error term d(κ) is a natural and useful strengthening of Howard's framework, and the conditional characterization of primitivity in Theorem B is an interesting converse direction. The paper is honest about its hypotheses and about which results are imported from Wei Zhang, Castella-Wan, Cornut-Vatsal, Jetchev-Skinner-Wan, and Castella-Hsieh; I did not detect circular reasoning, since Theorem A does not assume Howard's conjecture and Theorem B states it explicitly as an assumption. The main weaknesses are that several load-bearing local congruences are asserted tersely and that the application of the formal theory to anticyclotomic twists is not fully checked in the text.

major comments (3)

[§3.2, Propositions 3.2.2 and 3.2.3] The implication 'κ primitive ⇒ κ^Hg primitive' is the crucial bridge in Theorem A, but it is established in two very short paragraphs. The displayed congruences are not derived, and the passage from nonvanishing of D0DmΦP[m] modulo p to a nonzero class in H^1_{FΛ(m)}(K,T/m) is not written out. There is also a concrete inaccuracy: in the inert ordinary case, reducing Φ=(p+1)^2−a_p^2 modulo p gives (1−a_p^2)D0DmP[m], not (a_p^2−1)D0DmP[m] as printed. The sign difference is a p-adic unit and hence harmless for the intended conclusion, but it shows that these computations are delicate. Because a single missing or incorrect p-factor in these congruences would break the chain κ primitive ⇒ κ^Hg primitive ⇒ Λ-primitive ⇒ equality in Howard's Main Conjecture, I ask the authors to give a complete derivation of all three congruences, including the use of D0σ_p=D0σ_p^*=D0, and to explain explicitly why the resulting Kummer class is nonzero modulo p.
[§2.4, Theorem 2.4.5] The equality in part (c) when κ is Λ-primitive is a central ingredient of Theorem 3.3.1 and hence of Theorem A. In the proof, the assertion 'Exactly as in [MR04, Lemma 5.3.20], we have κ(P)≠0 when κ is Λ-primitive' is imported without detail, and the subsequent boundedness of d(κ(Q)) as Q varies is the mechanism that upgrades divisibility to equality. Since Definition 2.4.2 gives nonzero images only in quotients of the form T/(P,m^k) and at finite levels L_j, the implication to a nonzero class in the P-completion used in the argument is not immediate. The authors should either prove the relevant analogue of [MR04, Lemma 5.3.20] in Howard's setting or give a precise reference with the hypotheses verified.
[§4.4 and §5.2, twisted Kolyvagin systems] Theorem B applies Theorem 2.3.6 to the twisted Kolyvagin system κ^χ, but the manuscript does not explicitly check that the Selmer triple for T⊗χ, with χ a nontrivial anticyclotomic character, satisfies the hypotheses (H.0)–(H.5) of Section 2. In particular, the symmetric τ-conjugate self-dual pairing required in (H.4) is not constructed for the twisted module; since the natural Tate pairing pairs T⊗χ with T⊗χ^{-1}, this verification is not automatic. This is load-bearing because Theorem 2.3.6 supplies both the rank-one statement and the error-term formula used in the proof of Theorem B. The authors should state the Selmer triple for T⊗χ explicitly and verify the axioms, or point to a precise reference where this verification is carried out.

minor comments (3)

[Throughout] The text contains many typos and formatting errors, including 'HOW ARD'S' and 'KOL YV AGIN' in the title, 'conjeturally', 'endormorphism', 'trivally', 'W e', 'Frobeniuses', and 'a explicit reciprocity law' in the abstract. An editorial pass is needed.
[§5.2, proof of Theorem B] In the analytic rank one branch of Theorem B, the notation 'z_χ := κ^χ_1' is reused with χ trivial, and the displayed application of Corollary 5.1.2 is only valid in the first branch. The text should separate the two cases more clearly, especially because Corollary 5.1.2 requires s>f_0, which fails for trivial χ.
[§3.2, Proposition 3.2.2] The sign discrepancy in the inert ordinary congruence should be corrected, and the statement should specify that the displayed scalars are only determined up to a p-adic unit, which is all that the primitivity argument needs.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof upgrades Howard's divisibility to equality via a genuinely new error-term formalism and a non-constructive primitivity implication, with no fitted parameters or definitional identification of output with input.

full rationale

The paper's central argument is self-contained in the sense relevant to circularity analysis. The error-term formula (Theorem 2.3.6, equation (2.3.b)) is proved from Howard's Kolyvagin-system axioms together with new core-vertex arguments (Corollary 2.2.13, Lemma 2.3.1), and d(kappa)=0 iff kappa is primitive is a proved equivalence (Proposition 2.3.3), not a definition. The bridge from finite-level primitivity of kappa to Iwasawa-level primitivity of kappa^Hg (Propositions 3.2.2 and 3.2.3) is a congruence computation relating Q[m] to Phi P[m] modulo p; even if the displayed scalars were questionable, that would be a correctness risk, not circularity, because no term in the congruences is the conclusion being proved. Theorem 2.4.5(c) upgrades divisibility to equality on the genuinely additional hypothesis that kappa^Hg is Lambda-primitive, and Proposition 2.4.4 proves that finite-level primitivity implies Lambda-primitivity by a standard core-vertex argument. The cited results (Howard, MR, Castella-Wan, BCK, CH, JSW, Zhang) are external theorems with assumptions that do not include the paper's conclusion; they are not self-citations of the present author, and they are not used as a substitute for the actual implication proved here. Theorem B is explicitly conditional on Howard's Main Conjecture and uses it to derive an equivalence about primitivity and Tamagawa factors, which is a legitimate conditional statement, not a derivation that assumes its own conclusion. No fitted parameters, no renaming of a known result under new coordinates, and no uniqueness theorem imported from the same authors appear. The tersely stated congruences in Section 3.2 may deserve scrutiny for correctness, but they are not circular.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No numerical constants are fitted and no new entities are introduced. The paper's central results rest on the standard hypotheses above plus external deep theorems (Howard's formalism, Cornut-Vatsal, Wei Zhang, Jetchev-Skinner-Wan, Castella-Hsieh) used as black boxes. The only novel invariant is the formal error term d(kappa), a function of the Kolyvagin system, not a fitted parameter.

assumptions (8)

domain assumption Generalized Heegner Hypothesis (Heeg): N^- is square-free and has an even number of prime factors.
Standing hypothesis throughout; ensures existence of Heegner points and the Kolyvagin system.
domain assumption Standard local and global hypotheses (good), (p big), (res-surj): good reduction at p, p at least 5, surjective residual representation.
Needed to construct the Tate module Kolyvagin systems and to control Selmer groups.
domain assumption (not anom): p does not divide the size of the reduced elliptic curve at places above p.
Crucial in Propositions 3.1.3, 3.2.2 and 3.2.3 to guarantee p-unit multiples in the mod-p comparisons.
domain assumption (split): p splits in K.
Required for the supersingular construction in Section 3.2 and for the BDP and control theorems in Sections 4 and 5.
domain assumption Assumption 1.1.1: either N^- equals 1 or p does not divide the Tamagawa factor c_w(E/Q) for all w dividing N^+.
Ensures the classes kappa_n and their twists lie in the appropriate Selmer groups; the paper notes in Remark 1.1.2 that it can likely be removed.
domain assumption Assumption 1.1.7: (Heeg), (good), (split), (p big), (res-surj) plus either p does not divide ap(ap minus 1) or E/K has analytic rank 1.
Used in Theorem B to guarantee non-torsion localization of twisted Kolyvagin classes and the ordinary and non-anomalous local condition.
domain assumption Howard's Main Conjecture is assumed in Theorem B.
Theorem B is a conditional equivalence: it assumes the main conjecture to derive the BDP main conjecture via [BCK18, Theorem 4.1].
domain assumption Wei Zhang's Hypothesis in Theorem 1.1.5, a ramification condition on E[p].
Invoked to obtain primitivity of kappa from [Zha14] and hence new unconditional cases of the main conjecture.

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Pith. "Pith review of On Howard's main conjecture and the Heegner point Kolyvagin system." pith.science (2026). https://pith.science/paper/HXUMOCKL

@misc{pith2026190809197,
  author       = {Pith},
  title        = {Pith review of: On Howard's main conjecture and the Heegner point Kolyvagin system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HXUMOCKL}},
  note         = {Machine review of arXiv:1908.09197}
}

read the original abstract

We upgrade Howard's divisibility toward Perrin-Riou's Heegner point Main Conjecture to an equality under some mild conditions. We do this by exploiting Wei Zhang's proof of the Kolyvagin conjecture. The main ingredient is an improvement of Howard's Kolyvagin system formalism. As another consequence of it, we establish the equivalence between this main conjecture and the primitivity of the Kolyvagin system in certain cases, by also exploiting a explicit reciprocity law for Heegner points.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Indivisibility of Heegner points and arithmetic applications

Ashay Burungale, Francesc Castella, and Chan-Ho Kim. Indivisibility of H eegner points and arithmetic applications, 2018. arXiv:1806.01691

work page Pith review arXiv 2018

[2] [2]

Generalized H eegner cycles and p -adic R ankin L -series

Massimo Bertolini, Henri Darmon, and Kartik Prasanna. Generalized H eegner cycles and p -adic R ankin L -series. Duke Math. J. , 162(6):1033--1148, 2013. With an appendix by Brian Conrad

work page 2013

[3] [3]

L -functions and T amagawa numbers of motives

Spencer Bloch and Kazuya Kato. L -functions and T amagawa numbers of motives. In The G rothendieck F estschrift, V ol. I , volume 86 of Progr. Math. , pages 333--400. Birkh\" a user Boston, Boston, MA, 1990

1990

[4] [4]

Burungale

Ashay A. Burungale. On the non-triviality of the p -adic A bel- J acobi image of generalised H eegner cycles modulo p , II : S himura curves. J. Inst. Math. Jussieu , 16(1):189--222, 2017

2017

[5] [5]

Heegner cycles and p -adic L -functions

Francesc Castella and Ming-Lun Hsieh. Heegner cycles and p -adic L -functions. Math. Ann. , 370(1-2):567--628, 2018

work page 2018

[6] [6]

Nontriviality of R ankin- S elberg L -functions and CM points

Christophe Cornut and Vinayak Vatsal. Nontriviality of R ankin- S elberg L -functions and CM points. In L -functions and G alois representations , volume 320 of London Math. Soc. Lecture Note Ser. , pages 121--186. Cambridge Univ. Press, Cambridge, 2007

work page 2007

[7] [7]

Perrin-Riou's main conjecture for elliptic curves at supersingular primes

Francesc Castella and Xin Wan. Perrin- R iou's main conjecture for elliptic curves at supersingular primes, 2016. arXiv:1607.02019

work page Pith review arXiv 2016

[8] [8]

Benedict H. Gross. Kolyvagin's work on modular elliptic curves. In L -functions and arithmetic ( D urham, 1989) , volume 153 of London Math. Soc. Lecture Note Ser. , pages 235--256. Cambridge Univ. Press, Cambridge, 1991

work page 1989