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Kolyvagin's conjecture for modular forms

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Pith's one-line read This paper proves a higher-weight analogue of Kolyvagin's conjecture, showing that the strict Kolyvagin system of derived Heegner-cycle classes is nonzero for every admissible set, under mild technical assumptions.

desk verdict Genuine proof of the strong Kolyvagin conjecture for higher-weight modular forms in the ordinary p>k+1 range, conditional on a residual irreducibility assumption that the paper does not verify explicitly. read the letter →

arxiv 2412.02303 v2 pith:PCVRR27J submitted 2024-12-03 math.NT math.AG

Matteo Longo , Maria Rosaria Pati , Stefano Vigni This is my paper · ORCID
classification math.NTmath.AG MSC 11F1114C15
keywords Kolyvagin'sconjectureHeegnercyclesmodularformsKuga–SatovarietiesSelmergroupsTamagawanumberIwasawatheoryp-adicGaloisrepresentations
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 higher-weight analogue of Kolyvagin's conjecture: for a newform $f$ of even weight $k \geq 4$ and an ordinary prime $p > k+1$, the strict Kolyvagin system built from derived Heegner-cycle classes is nonzero for every admissible set $S$ in the indefinite case, conditional on a residual-irreducibility assumption. Kolyvagin's original conjecture predicted the $p$-indivisibility of derived Heegner points on elliptic curves; the present work extends the weight-two congruence method to Heegner cycles of higher weight. If correct, the main theorem yields the $p$-part of the Tamagawa number conjecture for the motive of $f$ when its analytic rank is $1$, together with structure theorems for Bloch–Kato–Selmer groups, a $p$-parity formula, and $p$-converse theorems.

What carries the argument

The load-bearing machinery is the strict Kolyvagin system $\kappa_S^\star = \{c_n(S)\}$: global classes in $H^1(K, T^\dagger_{f,p})$ obtained by applying Kolyvagin derivative operators $D_n$ to Heegner-cycle Abel–Jacobi images $y_n(S)$ and then descending from ring-class fields. The proof proceeds by 'triangulation' of Selmer groups: an induction that adds admissible primes to shrink the relevant Selmer group, and uses the two explicit reciprocity laws from [56] to transfer nonvanishing between finite and singular local conditions. A Selmer-rank-one theorem from [56] gives the base case, and a cyclotomic Iwasawa main conjecture result from [53] supplies the parity input that makes the Selmer dimension odd.

What would settle it

Take an explicit weight-4 newform $f$ with rational coefficients, an ordinary prime $p > 5$ satisfying Assumption 2.4, an imaginary quadratic field $K$, and an admissible prime $\ell$. If the corresponding $\theta$ element (the $p$-adic avatar of the central $L$-value) is not a $p$-adic unit, the second reciprocity law makes $\operatorname{loc}_\ell(c_1(S)) = 0$ in Selmer rank 1, contradicting the main theorem; equivalently, a numerical check that $\operatorname{loc}_\ell(c_1(S)) \neq 0$ for one concrete triple $(f,p,K,\ell)$ would exhibit the theorem's mechanism at work.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3.17: for every $S \in \mathcal{P}_{\mathrm{indef}}$, the strict Kolyvagin system $\kappa_S^\star$ is nonzero. In classical form, Kolyvagin's conjecture says the full system $\kappa_S$ contains a nonzero class; the strong form says the mod-$p$ classes $c_n(S)$ already do. The theorem is proved under Assumption 2.4 and $p > k+1$, and the proof constructs the systems from $\varphi$-isotypic Abel–Jacobi images of Heegner cycles on Kuga–Sato varieties over Shimura curves, with the classes $c_n(S)$ obtained by applying Kolyvagin derivative operators and descending from ring-class fields.

Load-bearing premise

The load-bearing premise is Assumption 2.4(2): the mod-$p$ Galois representation attached to $f$ must remain absolutely irreducible after restriction to $\mathbb{Q}(\sqrt{p^*})$; if it fails, the uniqueness of the representation, the Selmer-group comparisons, and the Iwasawa-theoretic input used in the induction lose control, and the paper verifies this condition for no explicit newform.

Editorial extensions

If this is right

  • For every admissible $S \in \mathcal{P}_{\mathrm{indef}}$, the strict system $\kappa_S^\star$ contains a nonzero class $c_n(S)$, so Kolyvagin's conjecture holds in strong form for higher-weight newforms under the stated assumptions.
  • The $p$-part of the Tamagawa number conjecture for the motive of $f$ holds whenever the analytic rank is 1 and the regulator and Abel–Jacobi assumptions in §4.2.1 are satisfied: algebraic rank equals analytic rank and the Bloch–Kato and Nekovář Tate–Shafarevich groups agree.
  • The structure of Bloch–Kato–Selmer groups of $f$ over $K$ is pinned down: the $\epsilon_\infty$-eigenspace has corank $\nu_\infty + 1$, the opposite eigenspace has corank at most $\nu_\infty$, and certain finite invariants satisfy the explicit divisibility relations of Theorem 5.1.
  • The $p$-parity formula $(-1)^{r_p(f)} = \epsilon(f)$ holds, and if the Selmer corank is 1 then the analytic rank is 1, giving a $p$-converse theorem for modular forms of higher weight.

Reading between the lines

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

  • The same strategy should combine with the Hida-family proof from [39] to cover, for a fixed $f$, all ordinary primes except possibly a finite exceptional set: the present paper requires $p > k+1$, while the Hida method applies when $k \equiv 2 \pmod{2(p-1)}$, forcing $p < k$.
  • A concrete test would be to verify Assumption 2.4(2) for a specific weight-4 newform; since the paper exhibits no example, the true scope of Theorem 3.17 is not yet known from the manuscript alone.
  • The authors' announced Heegner-cycle main conjecture would give a Perrin-Riou-style counterpart for higher weight and is a natural next target: if nontriviality of $\kappa_S^\star$ is the input, the main conjecture would then determine the Selmer group structure explicitly.
  • Extending the Selmer-rank-one theorem from [56] to non-ordinary primes would remove the ordinariness restriction from the induction; the authors note this is expected from ongoing work.
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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

2 major / 5 minor

Summary. The paper proves, under Assumption 2.4 and the standing hypothesis p>k+1, Kolyvagin's conjecture in its strong form for the strict Kolyvagin system of derived Heegner-cycle classes attached to an even-weight newform f of weight k≥4: for every admissible square-free S in the indefinite case, κ⋆_S ≠ {0} (Theorem 3.17). The strategy follows W. Zhang's congruence method in weight 2: level raising (Theorem 2.7), explicit reciprocity laws of Wang (Theorems 2.16 and 2.17), triangulation of Selmer groups (Proposition 3.14), a rank-one base case from Wang (Theorem 3.16), and parity from Skinner–Urban (Theorem 3.15). The paper also states applications to the p-part of the Tamagawa number conjecture (Theorem 4.2) and to structure theorems for Selmer groups, p-parity results, and p-converse theorems, with proofs deferred to the authors' earlier work [39].

Significance. If the hypotheses are satisfied, this is a substantial contribution: it extends Kolyvagin's conjecture from elliptic curves to higher-weight modular forms in the ordinary, p>k+1 range, complementing the Hida-theoretic approach of [39] in the p<k range. The core induction is carefully structured, and the paper is transparent about its conditionalities, citing Wang's indivisibility theorem and Skinner–Urban's nonvanishing results for the rank-one base and the parity input. The main unresolved point is that Assumption 2.4(2) is load-bearing for the Selmer comparison and the base case, yet no instance of a newform satisfying it is exhibited; the unconditional reach of the theorem is therefore unclear.

major comments (2)
  1. [§2.4, Assumption 2.4(2)] Assumption 2.4(2) — absolute irreducibility of the residual representation restricted to Gal(Qbar/Q(√p*)) — is load-bearing: it is used to ensure the uniqueness of the level-raised Galois representation (via [10, Théorème 3]), to justify the Selmer comparison in Lemma 2.9, and in the proof of Wang's base case Theorem 3.16 (Step 2) to pass from Selmer length to the valuation of the algebraic L-value. The paper neither exhibits a newform f for which this condition is verified nor argues that it follows from the other parts of Assumption 2.4 or holds generically. As a consequence, Theorem 3.17 is conditional on a hypothesis whose non-emptiness is not established. The authors should either provide a family of examples, give a genericity argument, or at least discuss the status of this condition and its relation to the assumptions in [56] and [57].
  2. [§2.6.6, Lemma 2.9] The comparison Sel_S(K, A†_{φ,℘}[℘]) ⊗ k_{℘S} ≃ Sel(K, A†_{φS,℘S}[℘S]) is proved in two sentences. The nontrivial point is the identification of local conditions at the primes dividing S for the original and level-raised representations; the proof invokes (2.9) and the module isomorphism A_{φS}[℘S] ≃ A_{φ,℘}⊗k_{℘S}, but the details of the local comparison are not shown. Since this lemma is used in Theorem 3.15 (to transfer nonvanishing from Sel(K,A_{f_T}) to Sel_T(K,T†_{f,p})) and in Theorem 3.16 (Step 2, in the length computation), the proof should be completed or the reader should be referred to a precise statement in [56] that covers this comparison.
minor comments (5)
  1. [§3.2.2] The sentence 'for each integer m with 1≤m≤n there is a natural Galois-equivariant injection' is unclear: the bound should presumably be 'for all m≥1' or '1≤m≤M(n)', matching the later use of the Kolyvagin index.
  2. [Proposition 3.14] In the statement, 'Sel_S(K/T†_{f,p})^{-ǫ_S}' should be 'Sel_S(K, T†_{f,p})^{-ǫ_S}'; the current notation obscures that this is a Selmer group in H^1(K,T†_{f,p}).
  3. [§5.1] The sign ǫ∞ is defined twice: once in (5.1) and again two paragraphs later as 'let ǫ∞∈{±} be the sign from (5.1)'. The second definition should be removed.
  4. [§2.4] In Assumption 2.4(2) the module T†_{φ,℘} is the residual quotient T†_{φ,℘}/℘T†_{φ,℘}; a brief reminder of this notation would improve readability, as the overline is easily lost in the printed text.
  5. [References] References [39] and [54] are cited as 'submitted' or 'arXiv:...'; if final publication data are available, the references should be updated.

Circularity Check

1 steps flagged · score 2.0 of 10

Core Kolyvagin theorem is externally grounded via Wang and Skinner–Urban; only the application sections are delegated to the authors' own earlier preprint [39], yielding a minor self-citation burden.

  1. self citation load bearing [Section 4.2.3 and opening of Section 5]
    "Theorem 4.2 can be proved exactly as [39, Theorem 4.41], observing that [39, Theorem 3.35], which proves Kolyvagin's conjecture in higher weight under the congruence condition k ≡ 2 (mod 2(p−1)), must be replaced here by Theorem 3.17. ... Since proofs can be found in [39], here we content ourselves with describing the statements of these by-products of Theorem 3.17."

    The advertised applications (Theorem 4.2 and Theorems 5.1–5.4) are not proved in this paper; their proofs are asserted to be identical to those in the authors' own submitted preprint [39], with only Theorem 3.17 substituted for [39, Theorem 3.35]. Thus the validity of the application statements rests on the authors' prior self-citation rather than on a derivation contained in this paper. This does not affect the central Theorem 3.17, whose proof is carried out from external results of Wang, Skinner–Urban, Chida–Hsieh, and Zhang, so the circularity burden is confined to the application sections and is minor.

full rationale

No circular reduction was found in the proof of the main theorem. Theorem 3.17 proves the strong form of Kolyvagin's conjecture by induction: the base case r = 1 is Wang's Theorem 3.16; the parity input is Theorem 3.15, which uses Wang's level raising, the functional equation, and Skinner–Urban; and the induction step applies Proposition 3.14 only after the inductive hypothesis has already supplied the non-vanishing κ⋆_{Sℓ1ℓ2} ≠ 0. Proposition 3.14's hypothesis (κ⋆_S ≠ 0) is not the same as the theorem being proved at that stage, so there is no equation or assumption that is used to prove itself. The explicit reciprocity laws (Theorems 2.16–2.17) and the Selmer comparisons are quoted from Wang; Assumption 2.4 is a genuine hypothesis needed for the uniqueness of Galois representations and for Skinner–Urban input, but the paper does not verify it for any explicit newform, which is a limitation of applicability rather than a circular step. The only self-citation burden is in Sections 4 and 5, where proofs are delegated to the authors' own earlier manuscript [39]; this is a real reliance on prior work by the same authors, but it does not feed back into the independent proof of Theorem 3.17. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No constants are fitted to data; the central claim depends only on the structural hypotheses below and on established external theorems. The conjectural inputs enter the applications in Sections 4 and 5, not the proof of Theorem A.

assumptions (9)
  • domain assumption Assumption 2.4(1): φ is ordinary at p, i.e. φ(T_p) is a p-adic unit.
    Imposed to define ordinary Selmer conditions at v|p and to apply Wang's theorems; if p is non-ordinary the theorem is not claimed.
  • domain assumption Assumption 2.4(2): residual representation T†_{φ,℘} restricted to Gal(Qbar/Q(√p*)) is absolutely irreducible.
    Underlies uniqueness of Galois representations, comparison of Selmer groups, and application of Skinner-Urban; the weakest load-bearing residual condition.
  • domain assumption Assumption 2.4(3)-(5): residual representation ramified at primes q|M with q≡1 mod p, at q|D with q≡±1 mod p, and at some q|MD.
    Ramification hypotheses are used in level raising (Theorem 2.7) and in descent arguments.
  • domain assumption Hypothesis (H-p), §2.4: p > k+1.
    Needed for the pairing in §2.2.1 and for Wang's [56] results; excludes primes below k+2.
  • domain assumption §3.1: N squarefree, N^- product of an even number of primes, and K imaginary quadratic with primes dividing N^+ split and primes dividing N^- inert.
    Heegner and quaternionic hypotheses required for the indefinite Shimura curve setup; the theorem is stated under them.
  • standard math Wang's indivisibility theorem and explicit reciprocity laws [56, Theorems 2.9, 2.12, 3.4, 3.6, 4.5, 4.6].
    Provide the rank-one base case, level raising, and p-adic unit formulas on which Theorem 3.16 and the induction rest.
  • standard math Skinner-Urban Iwasawa main conjecture for GL2 [53].
    Used in Theorem 3.15 to obtain nonvanishing of Selmer groups and parity.
  • domain assumption For Section 4 and Theorem 4.2: integral p-adic regulator map is an isomorphism and p-adic Abel-Jacobi maps are injective on Heegner-type cycles (§4.2.1), plus the Beilinson-Deligne rationality and Gillet-Soule height nondegeneracy assumptions from [39].
    These are explicitly conjectural; Theorem 4.2 is conditional on them.
  • domain assumption For Section 5.4 and Theorem 5.4: nondegeneracy of Gillet-Soule height pairings on Heegner modules (GS).
    Needed for the p-converse theorem; the paper notes this is beyond current techniques.

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Pith. "Pith review of Kolyvagin's conjecture for modular forms." pith.science (2026). https://pith.science/paper/PCVRR27J

@misc{pith2026241202303,
  author       = {Pith},
  title        = {Pith review of: Kolyvagin's conjecture for modular forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCVRR27J}},
  note         = {Machine review of arXiv:2412.02303}
}
abstract

Our main result in this article is a proof (under mild technical assumptions) of an analogue for $p$-adic Galois representations attached to a newform $f$ of even weight $k\geq4$ of Kolyvagin's conjecture on the $p$-indivisibility of derived Heegner points on elliptic curves, where $p$ is a prime number that is ordinary for $f$. Our strategy, which is inspired by work of W. Zhang in weight $2$, is based on a variant for modular forms of the congruence method originally introduced by Bertolini-Darmon to prove one divisibility in the anticyclotomic Iwasawa main conjecture for rational elliptic curves. We adapt to higher (even) weight modular forms this approach via congruences, building crucially on results of Wang on the indivisibility of Heegner cycles over Shimura curves. Then we offer an application of our results on Kolyvagin's conjecture to the Tamagawa number conjecture for the motive of $f$ and describe other (standard) consequences on structure theorems for Bloch-Kato-Selmer groups, $p$-parity results and converse theorems for $f$. Since in the present paper we need $p>k+1$, our main theorem and its applications can be viewed as complementary to results obtained by the first and third authors in their article on the Tamagawa number conjecture for modular motives, where Kolyvagin's conjecture was proved (in a completely different way exploiting the arithmetic of Hida families) under the assumption that $k$ is congruent to $2$ modulo $2(p-1)$, which forces $p<k$. In forthcoming work, we will use results contained in this paper to prove (under analogous assumptions) the counterpart for an even weight newform $f$ of Perrin-Riou's Heegner point main conjecture for elliptic curves ("Heegner cycle main conjecture" for $f$).

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