{"id":"fc2774ad-7295-4ebf-adc7-546f4b55363e","arxiv_id":"2608.06879","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove an integral Iwasawa main conjecture (characteristic ideal equals p-adic L-function) for CM elliptic curves at ramified primes, the first in a setting with no trianguline geometric specializations.","lead":"A team of Iwasawa theorists proves a new 'main conjecture' for elliptic curves with complex multiplication at primes that ramify in the CM field, matching a p-adic L-function to a signed Selmer group. It is the first such theorem for a p-adic deformation with no trianguline geometric specializations, which means standard tools for these conjectures do not apply.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.6 hinges on the unpublished local sign decomposition [5]; the sign containment loc_p(z) ∈ H^1_ε (Prop. 4.6) is the most directly testable load-bearing input.","rationale":"I read the paper as a globalisation of the authors' local sign decomposition, with the main theorem conditional on the unpublished preprint [5]. The global reductions are clearly laid out and no internal inconsistency surfaced: Theorem 3.14 derives the elliptic-unit main conjecture from Rubin's published theorem, Proposition 4.10 derives the interpolation formula from Kato's reciprocity law, and the control theorems in Section 6 are standard. The genuine soft spot is the black-box local input, exactly as the reader's weakest_assumption states. The non-trianguline assertion in Section 1.1 is indeed unproved; I flag it for completeness, but it is not load-bearing for Theorem 1.7 because no step of the proof invokes it. The concrete test I propose would settle the most dangerous single sign assertion (Prop. 4.6) using only the paper's own proved reciprocity law, and if it passes the remaining issue is the freeness/interpolation of the local decomposition in [5], which requires a separate review of that preprint. Since the reader already made the verdict CONDITIONAL on the status of [5], my assessment does not change the verdict.","tokens_in":42044,"tokens_out":15695,"duration_ms":157840,"concrete_test":"Test the sign in Proposition 4.6 by combining the paper's own explicit reciprocity law (Proposition 4.10) with Rohrlich's nonvanishing theorem. Choose a finite-order anticyclotomic character χ with ε(φχ)=+1 and L_{pf}(φχ,1)≠0 (such χ exist by [28] and Lemma 4.4). Use (4.11) to compute exp^*(loc_p(z^{ac}_{p^∞ f})) at χ. If the value is nonzero, loc_p(z) cannot lie in H^1_{-ε}, because H^1_{-ε} forces the dual exponential to vanish at all χ with ε(φχ)=+1; this confirms the containment loc_p(z)∈H^1_ε used in Theorem 5.5 and (5.3). If the computed value vanishes, the sign convention imported from [5, Prop. 8.11] is wrong and Theorem 5.6 would fail as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the central theorem (Theorem 5.6, hence Theorem 1.7) reduces through the exact sequence (5.3) and the characteristic-ideal computation (5.4) to two assertions imported from the unpublished preprint [5]: (i) the local Rubin-type decomposition of Theorem 2.5, namely that H^1(Ψ,T_ψ) splits as free rank-one Lagrangian Λ-modules H^1_+ ⊕ H^1_- interpolating the Bloch–Kato subgroups via completed ε-constants; and (ii) Proposition 4.6, which places the anticyclotomic elliptic unit loc_p(z^{ac}_{p^∞ f}) in H^1_ε. If (i) fails, the signed Selmer groups, the Gaussian local points, and the p-adic L-function of Definition 4.7 have no basis, and the equality Ch_Λ(X^{-ε}) = (L_{p,v_ε}) in Theorem 5.6 is not even well posed. If (ii) has the wrong sign, the roles of X^ε and X^{-ε} in (5.3) are interchanged and the stated main conjecture is false. The paper gives no proof of either item, citing [5, Thm. 7.25] and [5, Prop. 8.11]; Assumption 2.1 is vacuous for p≥5 ramified in a class-number-one CM field, so this is not a mitigating hypothesis. Separately, Section 1.1 asserts without proof that no geometric specialisation is trianguline at p; this supports the novelty claim but is not used in the proof of Theorem 1.7.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops an integral Iwasawa theory for anticyclotomic Z_p-extensions of CM fields at primes p ramified in K. For a conjugate symplectic self-dual Hecke character φ of infinity type (1,0), the authors use the local sign decomposition of [5] to define signed Selmer groups Sel^{±}, a Rubin-type p-adic L-function L_{p,v_ε}(E) as the coordinate of the anticyclotomic elliptic unit in the ε-Lagrangian submodule, and prove an integral Iwasawa main conjecture Ch_Λ(X^{-ε}(E)) = (L_{p,v_ε}(E)) (Theorem 1.7). They also prove an interpolation formula for L_p(E) at de Rham specialisations with ε(φχ)=+1 (Theorem 1.5), and a formula at twists with ε(φχ)=−1 relating L_p(E) to logarithms of Selmer elements (Theorem 1.8). The main proofs rely on a descent from Rubin's elliptic unit main conjecture (Theorem 3.14), an explicit reciprocity law (Prop. 4.10), and the sign containment of the elliptic unit (Prop. 4.6).","tokens_in":42335,"tokens_out":5309,"duration_ms":50389,"significance":"If the local results imported from [5] are valid, the paper achieves a genuine milestone: the first Iwasawa main conjecture of p-adic-L-function type for a deformation with no trianguline geometric specialisation, together with a precise control theorem and asymptotic rank formula. The paper is carefully structured and proves substantial results in the text: the Gaussian plus/minus polynomials and their class-number interpretation (Lemma 2.11), the descent from Rubin's theorem to the anticyclotomic elliptic unit main conjecture (Theorem 3.14), and the explicit reciprocity law (Prop. 4.10). The central deficit is that Theorem 2.5 and Proposition 4.6, on which the entire framework is built, are imported from an unpublished preprint and asserted without self-contained proof.","major_comments":[{"comment":"The decomposition H^1(Ψ, T_ψ) = H^1_+ ⊕ H^1_- into free rank-one Lagrangian Λ-submodules is stated as a consequence of [5, Thm. 7.25] and Lemma 2.3, but no proof is given in this paper. This decomposition is the foundation for the signed Selmer groups (Definition 5.1), the local points (Theorem 2.18), the p-adic L-function (Definition 4.7), and the characteristic-ideal equality in Theorem 5.6. As [5] is an unpublished preprint by the same authors, the main theorem is conditional on an external result. Please either include a full proof of Theorem 2.5 or state the main conjecture explicitly as conditional on [5] being made publicly available and refereed.","section":"Section 2, Theorem 2.5"},{"comment":"The containment loc_p(z^{ac}_{p^∞ f}) ∈ H^1_ε is the sign input that selects the correct Selmer group in Theorem 1.7. The proof is deferred with 'one may proceed just as in the proof of [29, Cor. 3.3] or [5, Prop. 8.11]'. This step is directly load-bearing: if the sign were reversed, the roles of ε and −ε in the exact sequence (5.3) would be interchanged and the stated equality Ch_Λ(X^{-ε}) = (L_{p,v_ε}) would be false. The proof must be written out.","section":"Section 4, Proposition 4.6"}],"minor_comments":[{"comment":"The assertion that no geometric specialisation is trianguline at p is stated without proof or reference. Since this claim appears in the abstract and frames the novelty but is not used in the proof of Theorem 1.7, please add a proof or reference, or explicitly mark it as a conjecture or expectation.","section":"Section 1.1"},{"comment":"The sign convention for ϵ(Ψ) differs from the notation in [5], as noted in Remark 4.3 for the global case; a short table comparing the two conventions would help the reader avoid sign errors when checking the local inputs.","section":"Definition 2.9 and Remark 4.3"},{"comment":"The factor (−2πp/|d_K|)^k involves the notation 2πp which is only implicitly defined; please spell out that this is the p-adic period associated to the choices of e_{Q_p} and the Haar measures.","section":"Section 4.4.1, equation (4.11)"},{"comment":"Reference [32] is given as 'Ph.D. thesis, Princeton University' with no title, year, or other identifying information; please provide the full citation.","section":"References, [32]"},{"comment":"The running title in the text contains spacing artifacts ('IW ASA W A THEOR Y'); please ensure the compiled manuscript is free of such artifacts.","section":"Title and front matter"}],"recommendation":"major_revision","confidential_remarks":"The two major issues both stem from the same dependency: the local sign decomposition of [5] carries the entire weight of the construction, while the present manuscript supplies no proof of Theorem 2.5 or Proposition 4.6. I would recommend requesting the authors to include complete proofs of these two results, or to resubmit as a combined manuscript with [5] so that the referee can verify the logical dependency in full."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis is a serious paper. It delivers the first integral Iwasawa main conjecture for a p-adic deformation with no trianguline geometric specialisations, in the anticyclotomic CM setting at ramified primes. The signed Selmer groups, the p-adic L-function via the epsilon-Lagrangian, and the equality to the characteristic ideal are all new. The construction of plus/minus local points from Gaussian cyclotomic polynomials is a clever new tool; it actually recovers class number and fundamental unit factors. The interpolation formula for arbitrary infinity type is also new. The internal logic is transparent: the main conjecture is reduced to Rubin's elliptic-unit main conjecture together with local sign decomposition. There is no fitted-parameter circularity: L_p is defined as the coordinate of the anticyclotomic elliptic unit in the epsilon-Lagrangian, then proved to interpolate L-values and to match the characteristic ideal.\n\nThe soft spot is structural. Theorems 2.5 and Proposition 4.6, the local sign decomposition and the sign of the elliptic unit, are imported from the authors' preprint [5], cited as Theorem 7.25 and Prop 8.11 there. The paper gives no proof of either. This is not a cosmetic dependency: if the local decomposition fails, the signed Selmer groups, L_p, and the main conjecture are not well-defined. If the sign in Prop 4.6 is wrong, the main conjecture as stated would be false. The stress-test note is right about both points. The authors are open about the dependency, and [5] is from a group with a strong record (their Rubin conjecture paper at inert primes is in Annals). Still, a referee cannot verify this paper without verifying [5]. I would suggest the journal require the authors to make the preprint available in final form, or to include the local results as an appendix if space allows.\n\nThe assertion that none of the geometric specializations is trianguline at p is also stated without proof in the introduction and abstract. It is not used in the proof of Theorem 1.7, but it is part of how the paper positions itself. That should be a short lemma or a precise citation. Minor.\n\nOverall: yes, send to a serious referee. The paper is for Iwasawa theorists, and it deserves a careful vetting of the local input. My recommendation: accept-shape after the referee checks [5] and the authors either prove or accurately cite the non-trianguline claim.","headline":"A real main conjecture at ramified primes, but the load-bearing local input is an unpublished prequel; referee both together.","tokens_in":42907,"tokens_out":3380,"would_cite":true,"duration_ms":35567,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R23","11G05","11G15","11S40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves an integral Iwasawa main conjecture for CM elliptic curves at ramified primes: the characteristic ideal of the signed Selmer group $X^{-\\varepsilon}(E)$ equals the ideal of the $p$-adic $L$-function…","keywords":["Iwasawa main conjecture","CM elliptic curves","anticyclotomic Z_p-extension","ramified primes","signed Selmer groups","local sign decomposition","Gaussian cyclotomic polynomials","p-adic L-functions"],"falsifier":"Compute a single finite layer: take a ramified quadratic extension $\\Psi/\\mathbb{Q}_p$ and a character $\\psi$ satisfying the conjugate symplectic self-dual condition, and check whether the plus/minus Bloch–Kato subgroups really give the claimed $p$-torsion-free direct-sum decomposition of Theorem 2.18; a failure at any $n$ would contradict the construction. Alternatively, find one anticyclotomic character $\\chi$ whose local epsilon constant does not follow the quadratic-residue rule $\\varepsilon(\\phi\\chi^b) = (b/p)\\varepsilon(\\phi\\chi)$, which would break the equidistribution underlying the signed framework.","tokens_in":41816,"feed_emoji":"🧮","tokens_out":6764,"duration_ms":65656,"temperature":0.7,"pith_summary":"This paper aims to establish an integral Iwasawa main conjecture for CM elliptic curves at primes $p$ ramified in the imaginary quadratic field, a regime where earlier frameworks stop working. The central result equates the characteristic ideal of a signed anticyclotomic Selmer group, built from one half of a canonical splitting of local Iwasawa cohomology, with the ideal generated by the $p$-adic $L$-function. The root numbers of twists split evenly at every layer of the anticyclotomic tower, so no single local condition interpolates the Bloch–Kato subgroups; the signed framework handles both signs at once. If correct, this gives the first main conjecture for a $p$-adic deformation none of whose geometric specialisations is trianguline at $p$, and it ties the $p$-adic $L$-function both to central $L$-values and to Selmer classes responsible for Mordell–Weil growth.","feed_headline":"Signed Selmer group equals p-adic L-function at ramified primes","feed_subtitle":"First main conjecture for a deformation with no trianguline specialisation; root-number signs split evenly per layer.","key_machinery":"The load-bearing object is the local sign decomposition from the prequel: for the conjugate symplectic self-dual deformation over the ramified local field, the local Iwasawa cohomology $H^1(\\Psi, T_\\psi)$ splits as a direct sum of two free rank-one Lagrangian $\\Lambda$-submodules $H^1_+ \\oplus H^1_-$ whose specialisations at de Rham twists reproduce the Bloch–Kato subgroups exactly when the completed local $\\varepsilon$-constant has the matching sign. On top of this, the paper introduces Gaussian plus/minus cyclotomic polynomials $\\Phi^\\pm_k(\\gamma)$, products over all anticyclotomic characters whose $\\varepsilon$-constant has a given sign; these polynomials annihilate the opposite-sign specialisations and generate the signed Bloch–Kato subgroups at finite layers. The signed Selmer groups are then defined by the corresponding Lagrangian local conditions, and the proof of the main conjecture globalises them via elliptic units: an anticyclotomic elliptic-unit main conjecture, descended from the classical elliptic-unit main conjecture, is transformed by Poitou–Tate duality into the equality of characteristic ideals.","core_discovery":"The authors' central claim, stated as Theorem 1.7, is that for a CM elliptic curve $E$ over $\\mathbb{Q}$, an odd prime $p \\geq 5$ ramified in the CM field, and the sign $\\varepsilon$ determined by the ratio of global and local epsilon constants, the signed Selmer group $X^{-\\varepsilon}(E)$ is a torsion module over the anticyclotomic Iwasawa algebra $\\Lambda$, and its characteristic ideal is exactly the principal ideal generated by the integral $p$-adic $L$-function: $\\mathrm{Ch}_\\Lambda(X^{-\\varepsilon}(E)) = (\\mathscr{L}_{p,v_\\varepsilon}(E))$ as ideals of $\\Lambda$. This is complemented by an interpolation theorem showing that $\\mathscr{L}_p(E)$ recovers the central Hecke $L$-values of the twists with epsilon-constant $+1$, for characters of arbitrary infinity type, and by a theorem expressing its values at twists with epsilon-constant $-1$ as a product of a dual exponential value and a Bloch–Kato logarithm of a Selmer class. The paper also proves that the Mordell–Weil rank over the $n$-th layer satisfies $\\operatorname{rank}_{\\mathcal{O}_K} E(K^{\\mathrm{ac}}_n) = (p^n-1)/2 + c$ for all sufficiently large $n$.","pith_inferences":["If the same two-Lagrangian template is applied to other deformations whose local epsilon constants equidistribute, one would predict analogous signed main conjectures, with the Gaussian polynomials replaced by finer sign filtrations.","The characteristic-ideal equality, combined with standard control theorems, should determine the asymptotic order of the $p$-primary Tate–Shafarevich groups at finite layers in terms of special values of $\\mathscr{L}_p$; this is a concrete numerical prediction.","The formula at $\\varepsilon = -1$ twists suggests a derivative-type relation: the first Taylor coefficient of $\\mathscr{L}_p$ at such a twist should be the height or logarithm of the corresponding Selmer class, analogous to BDP-type formulas in the inert-prime setting.","The plus/minus local points constructed here are natural candidates for the local input in a ramified-prime Euler system, which would give an independent route to the main conjecture."],"forward_implications":["The $p$-adic $L$-function $\\mathscr{L}_{p,v_\\varepsilon}(E)$ determines the full characteristic ideal of the signed Selmer group $X^{-\\varepsilon}(E)$, so the algebraic structure of that Selmer group is governed by an analytic object.","The interpolation formula recovers central Hecke $L$-values at all de Rham twists with $\\varepsilon = +1$, including twists of arbitrary infinity type, making $\\mathscr{L}_p(E)$ a genuine $p$-adic $L$-function for the ramified anticyclotomic deformation.","Values of $\\mathscr{L}_p(E)$ at twists with $\\varepsilon = -1$ are identified with Bloch–Kato logarithms and dual exponentials of Selmer classes, linking the $L$-function outside its interpolation range to non-torsion cohomology classes.","The control theorem yields the asymptotic formula $\\operatorname{rank}_{\\mathcal{O}_K} E(K^{\\mathrm{ac}}_n) = (p^n-1)/2 + c$ for $n \\gg 0$, matching the equidistribution of root numbers.","The theorem supplies a test case in which a main conjecture is provable even though no geometric specialisation of the deformation is trianguline at $p$."],"supporting_citations":[{"why":"Supplies the local sign decomposition, the Rubin-type conjecture resolving the decomposition of $H^1(\\Psi,T_\\psi)$ into Lagrangian submodules, and the construction of $\\mathscr{L}_p(E)$.","marker":"[5]"},{"why":"Provides the elliptic-unit main conjecture for imaginary quadratic fields, from which the authors descend their anticyclotomic elliptic-unit main conjecture.","marker":"[30]"},{"why":"Supplies the elliptic-unit framework and explicit reciprocity laws used in the interpolation formula and the local pairing computation.","marker":"[18]"},{"why":"Provides the Euler-system machinery establishing torsionness of Selmer groups and the characteristic-ideal inclusion used in the proof.","marker":"[31]"},{"why":"Gives the anticyclotomic non-vanishing of Hecke $L$-values used to prove $\\mathscr{L}_p(E)$ is non-zero and to drive the Mordell–Weil rank formula.","marker":"[28]"},{"why":"Supplies the plus/minus Iwasawa-theory template, including signed Selmer groups and plus/minus local conditions.","marker":"[19]"},{"why":"Provides the rational main-conjecture descent over $\\mathbb{Q}_p$ used in the elliptic-unit main conjecture.","marker":"[17]"}],"fun_headline_variants":["Signed Selmer = p-adic L-function at ramified primes","First main conjecture for non-trianguline deformations","Equality of Selmer and L-function for CM curves at ramified primes","Main conjecture proven for CM curves at ramified primes","Ramified primes: signed Selmer equals L-function"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction rests on the prequel's local sign decomposition, the claim that over each layer of the ramified anticyclotomic tower the local deformation splits into two rank-one Lagrangian pieces that exactly track the Bloch–Kato subgroups by sign; if that decomposition fails, the signed Selmer groups, the $p$-adic $L$-function, and the main conjecture all collapse.","fun_headline_variants_meta":{"raw":{"variants":["Signed Selmer = p-adic L-function at ramified primes","First main conjecture for non-trianguline deformations","Equality of Selmer and L-function for CM curves at ramified primes","Main conjecture proven for CM curves at ramified primes","Ramified primes: signed Selmer equals L-function"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001046,"raw_usage":{"total_tokens":4493,"prompt_tokens":1137,"completion_tokens":3356,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":753,"completion_tokens_details":{"reasoning_tokens":3271}},"tokens_in":753,"tokens_out":3356,"duration_ms":23092,"temperature":1.0,"reasoning_tokens":3271,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:01:28.905788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a single finite layer: take a ramified quadratic extension $\\Psi/\\mathbb{Q}_p$ and a character $\\psi$ satisfying the conjugate symplectic self-dual condition, and check whether the plus/minus Bloch–Kato subgroups really give the claimed $p$-torsion-free direct-sum decomposition of Theorem 2.18; a failure at any $n$ would contradict the construction. Alternatively, find one anticyclotomic character $\\chi$ whose local epsilon constant does not follow the quadratic-residue rule $\\varepsilon(\\phi\\chi^b) = (b/p)\\varepsilon(\\phi\\chi)$, which would break the equidistribution underlying the signed framework.","supporting_citations":[{"cited_title":"A local sign decomposition for symplectic self-dual Galois representations of rank two","cited_arxiv_id":"2508.17776","evidence_quote":"Supplies the local sign decomposition, the Rubin-type conjecture resolving the decomposition of $H^1(\\Psi,T_\\psi)$ into Lagrangian submodules, and the construction of $\\mathscr{L}_p(E)$."},{"cited_title":"main conjectures","cited_arxiv_id":null,"evidence_quote":"Provides the elliptic-unit main conjecture for imaginary quadratic fields, from which the authors descend their anticyclotomic elliptic-unit main conjecture."},{"cited_title":"Kato,p-adic Hodge theory and values of zeta functions of modular forms, Ast´ erisque295(2004), 117–290","cited_arxiv_id":null,"evidence_quote":"Supplies the elliptic-unit framework and explicit reciprocity laws used in the interpolation formula and the local pairing computation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Euler-system machinery establishing torsionness of Selmer groups and the characteristic-ideal inclusion used in the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the anticyclotomic non-vanishing of Hecke $L$-values used to prove $\\mathscr{L}_p(E)$ is non-zero and to drive the Mordell–Weil rank formula."},{"cited_title":"Kobayashi,Iwasawa theory for elliptic curves at supersingular primes, Invent","cited_arxiv_id":null,"evidence_quote":"Supplies the plus/minus Iwasawa-theory template, including signed Selmer groups and plus/minus local conditions."},{"cited_title":"Johnson-Leung and G","cited_arxiv_id":null,"evidence_quote":"Provides the rational main-conjecture descent over $\\mathbb{Q}_p$ used in the elliptic-unit main conjecture."}],"review_version":1}