{"id":"15be72e9-81f3-4220-8ba5-435eb80f2230","arxiv_id":"2412.02303","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under ordinary p and p>k+1 with standard residual assumptions, the strong Kolyvagin conjecture for even-weight newforms holds: derived Heegner-cycle cohomology classes are nonzero.","lead":"This paper proves a higher-weight analogue of Kolyvagin's conjecture for modular forms: for an even-weight newform and a suitable ordinary prime p, Kolyvagin classes built from Heegner cycles are all nonzero. It also draws consequences for the Tamagawa number conjecture, Selmer group structure, parity, and converse theorems under standard conjectural inputs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 2.4(2) is the load-bearing condition: it is needed for the Selmer comparisons and Wang's base case, yet no explicit newform satisfying it is given.","rationale":"I read Theorem 3.17 as a conditional theorem: under Assumption 2.4 and p>k+1, the strict Kolyvagin system for every S in P_indef is nonzero. The proof is internally coherent: the base case is Wang's published result, the parity input is Skinner-Urban, and the triangulation argument follows Zhang's induction. I found no circular use of Kolyvagin's conjecture and no obvious false step in the dimension inequalities of Proposition 3.14 or Theorem 3.17. The weakest point is indeed Assumption 2.4(2). It is used in the comparison of Selmer groups (Lemma 2.9) and in Wang's base case, so if it fails the induction has no starting point and the level-raised representation is not controlled. The paper neither verifies the condition for a single newform nor shows it is automatic from the other assumptions. That does not make the conditional theorem false, but it is a genuine restriction on the scope of the result and explains the medium correctness risk. My proposed test would exhibit at least one concrete instance or reveal that the condition is more restrictive than the paper suggests; either outcome is informative for readers relying on the theorem.","tokens_in":36479,"tokens_out":19903,"duration_ms":231106,"concrete_test":"Take an explicit newform f satisfying the other hypotheses, e.g. a weight-4 newform of small square-free level N with an even number of inert primes in K and an ordinary prime p>k+1, and compute its mod p Galois representation via modular symbols or a database of modular forms. Then check whether the image of Gal(Qbar/Q(√p*)) is absolutely irreducible, for instance by verifying that the characteristic polynomials of Frobenius at a Chebotarev-generating set of primes do not all lie in a common Borel subgroup (equivalently, the projective image is not dihedral with quadratic field Q(√p*)). If such an f exists, the conditional theorem has a concrete instance; if Assumption 2.4(2) is verified there, the main remaining risk is the depth of the external inputs rather than an empty or inconsistent hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem (Thm 3.17) is conditional, and its most load-bearing hypothesis is Assumption 2.4(2): the residual representation Tbar†_{φ,℘} restricted to Gal(Qbar/Q(√p*)) is absolutely irreducible. This is not cosmetic. It is invoked, via [10, Thm 3] and Lemma 2.9, to ensure that the level-raised eigenform φS from Thm 2.7 has a well-defined, unique Galois representation and that the Selmer group SelS(K,Tbar†_{φ,p}) can be identified with Sel(K,Tbar†_{φS,pS}); the same irreducibility is used in the proof of Wang's Thm 3.16 (Step 2) to pass from Selmer length to the valuation of the algebraic L-value. If Assumption 2.4(2) fails, the dimension computations in Proposition 3.14 and the induction in Thm 3.17 lose their control, and the base case of the induction has no justification. The paper gives no explicit newform f for which 2.4(2) is checked, nor does it prove that 2.4(2) follows from the other parts of Assumption 2.4; in particular, the theorem has no apparent unconditional instance. This is a limitation of applicability rather than an internal contradiction, which is why the verdict remains conditional.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":36754,"tokens_out":13696,"duration_ms":132149,"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":[{"comment":"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].","section":"§2.4, Assumption 2.4(2)"},{"comment":"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.","section":"§2.6.6, Lemma 2.9"}],"minor_comments":[{"comment":"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.","section":"§3.2.2"},{"comment":"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}).","section":"Proposition 3.14"},{"comment":"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.","section":"§5.1"},{"comment":"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.","section":"§2.4"},{"comment":"References [39] and [54] are cited as 'submitted' or 'arXiv:...'; if final publication data are available, the references should be updated.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is largely a well-structured conditional proof, but the main theorem is only as strong as Assumption 2.4(2), and the authors do not indicate whether any concrete newform satisfies it. I would not reject on these grounds, since the issue is fixable by an added discussion or a family of examples, but it should be addressed before the paper can be accepted. The applications in Sections 4 and 5 are entirely delegated to [39], so the novelty of the paper rests on Theorem 3.17; the reader should be told clearly that those sections contain no new proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuine proof of a theorem, not a conditional sketch. The paper proves the strong Kolyvagin conjecture for every admissible S in the indefinite case, for even weight k≥4 and ordinary p>k+1, under Assumption 2.4. The method is Zhang's congruence approach adapted to higher weight; the rank-1 base is Wang's theorem, parity comes from Skinner-Urban, and the triangulation induction is written out carefully. That the result is complementary to the authors' earlier [39] is stated honestly, and the applications are flagged as contingent on standard conjectures.\n\nThe proof is careful. Proposition 3.14 is the heart, and the inductive step in Theorem 3.17 is structured properly. The reliance on Wang, Skinner-Urban, and the reciprocity laws is explicit; I found no circular use of Kolyvagin's conjecture. Sections 4 and 5 delegate most proofs to [39], which is acceptable if a referee checks that the hypotheses align.\n\nThe soft spot is exactly what the stress-test flags: Assumption 2.4(2) is load-bearing but not verified for any explicit f. It is used in the Selmer comparison and in the proof of Wang's Theorem 3.16, so the induction has no base without it. The paper does not show that (2) follows from the rest of Assumption 2.4, and it gives no example. That makes the theorem conditional in a way that limits unconditional applicability, but it is not a contradiction in the argument. A short discussion of how restrictive (2) is, or a family of examples, would strengthen the paper.\n\nThe external dependencies are deep and not independently verified; that is normal in this area. The paper is not fully self-contained, but that is not a defect.\n\nThis paper deserves a serious referee. The target audience is Iwasawa theorists and arithmetic geometers working on modular forms. I would send it out, asking the authors to clarify the status of Assumption 2.4(2) and to double-check compatibility with Wang's hypotheses. If the external results hold, the paper is a solid contribution.","headline":"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.","tokens_in":37331,"tokens_out":3831,"would_cite":true,"duration_ms":38204,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","14C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Kolyvagin's conjecture","Heegner cycles","modular forms","Kuga–Sato varieties","Selmer groups","Tamagawa number conjecture","Iwasawa theory","p-adic Galois representations"],"falsifier":"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.","tokens_in":36269,"feed_emoji":"🔢","tokens_out":10205,"duration_ms":96059,"temperature":0.7,"pith_summary":"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.","feed_headline":"Kolyvagin's conjecture proved for even-weight modular forms","feed_subtitle":"Nonzero derived Heegner-cycle systems unlock the Tamagawa number conjecture's $p$-part in rank 1.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the two explicit reciprocity laws for Heegner cycles over Shimura curves and the Selmer-rank-1 indivisibility theorem that is the base case of the induction.","marker":"[56]"},{"why":"Provides the weight-2 triangulation strategy and level-raising argument that the paper adapts to higher weight.","marker":"[57]"},{"why":"Introduces the congruence method and admissible-prime setup for one divisibility in the anticyclotomic Iwasawa main conjecture, which is the template here.","marker":"[3]"},{"why":"Gives the cyclotomic Iwasawa main conjecture input used to prove that Selmer dimensions are odd in Theorem 3.15.","marker":"[53]"},{"why":"Formulates the higher-weight Kolyvagin conjecture, defines the relevant Selmer groups, and provides auxiliary lemmas used in the triangulation.","marker":"[42]"},{"why":"Supplies the admissibility, theta-element, and ordinary-Selmer-group framework for modular forms of higher weight.","marker":"[17]"},{"why":"Gives the Kummer-type injections and vanishing lemma used to define the Kolyvagin classes $c_{n,m}(S)$ and to make the descent from ring-class fields valid.","marker":"[41]"},{"why":"Sets up the applications: the Tamagawa-number-conjecture statement for modular motives and the structure, parity, and converse theorems that Theorem 3.17 feeds into.","marker":"[39]"}],"fun_headline_variants":["Proof of Kolyvagin's conjecture for even-weight modular forms","Kolyvagin's conjecture proven via Heegner cycles on Shimura curves","Nonzero Kolyvagin systems settle conjecture for modular forms","Even-weight modular forms: Kolyvagin's conjecture proved"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Proof of Kolyvagin's conjecture for even-weight modular forms","Kolyvagin's conjecture proven via Heegner cycles on Shimura curves","Nonzero Kolyvagin systems settle conjecture for modular forms","Even-weight modular forms: Kolyvagin's conjecture proved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1550,"prompt_tokens":1064,"completion_tokens":486,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":680,"completion_tokens_details":{"reasoning_tokens":410}},"tokens_in":680,"tokens_out":486,"duration_ms":4856,"temperature":1.0,"reasoning_tokens":410,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:37:49.777690+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Wang, Indivisibility of Heegner cycles over Shimura curves and Se lmer groups , J","cited_arxiv_id":null,"evidence_quote":"Supplies the two explicit reciprocity laws for Heegner cycles over Shimura curves and the Selmer-rank-1 indivisibility theorem that is the base case of the induction."},{"cited_title":"Zhang, Selmer groups and the indivisibility of Heegner points , Camb","cited_arxiv_id":null,"evidence_quote":"Provides the weight-2 triangulation strategy and level-raising argument that the paper adapts to higher weight."},{"cited_title":"Bertolini and H","cited_arxiv_id":null,"evidence_quote":"Introduces the congruence method and admissible-prime setup for one divisibility in the anticyclotomic Iwasawa main conjecture, which is the template here."},{"cited_title":"Skinner and E","cited_arxiv_id":null,"evidence_quote":"Gives the cyclotomic Iwasawa main conjecture input used to prove that Selmer dimensions are odd in Theorem 3.15."},{"cited_title":"Masoero, On the structure of Selmer and Shafarevich–Tate groups of ev en weight modular forms , Trans","cited_arxiv_id":null,"evidence_quote":"Formulates the higher-weight Kolyvagin conjecture, defines the relevant Selmer groups, and provides auxiliary lemmas used in the triangulation."},{"cited_title":"Chida and M.-L","cited_arxiv_id":null,"evidence_quote":"Supplies the admissibility, theta-element, and ordinary-Selmer-group framework for modular forms of higher weight."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Kummer-type injections and vanishing lemma used to define the Kolyvagin classes $c_{n,m}(S)$ and to make the descent from ring-class fields valid."},{"cited_title":"The Tamagawa number conjecture and Kolyvagin's conjecture for motives of modular forms","cited_arxiv_id":"2211.04907","evidence_quote":"Sets up the applications: the Tamagawa-number-conjecture statement for modular motives and the structure, parity, and converse theorems that Theorem 3.17 feeds into."}],"review_version":1}