{"id":"1d22464c-f835-4dfb-8b38-14915b76d0ba","arxiv_id":"2509.07564","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Anticyclotomic diagonal cycle classes are shown to match Beilinson-Flach elements up to explicit factors for a CM weight-one Eisenstein degeneration.","lead":"This math paper shows that two different families of arithmetic cohomology classes, one tied to imaginary quadratic extensions and one tied to the cyclotomic direction, coincide for modular forms after multiplying by explicit p-adic L-values. The result extends previous comparisons and yields factorization formulas, but the main statement is conditional on unproved rank assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central comparison depends on an unproved rank-one/torsion-free Selmer hypothesis; if it fails, the injectivity step in Cor 5.19 collapses and Theorem 1.1 is not established.","rationale":"The paper's central claim is a comparison of two cohomology classes. The only step that promotes an equality of Perrin-Riou images (Theorem 5.18) to an equality of classes is the injectivity of the map L, which is made to depend on the Selmer group H^1_{G∪+}(Q,V^†_{fgh}) being torsion-free of rank 1. This is not proved; Remark 5.20 explicitly defers it to sign considerations. The reader's weakest_assumption identified precisely this hypothesis. I do not see an internal inconsistency: the algebra of p-adic L-functions in Proposition 5.2, the local computations in §5.4, and the reciprocity laws all appear coherent. The non-vanishing assumption on L^p_g is also stated. The exceptional-zero applications are further contingent on Conjecture 6.8, as the paper admits. Thus the result is a promising conditional theorem, exactly matching the reader's CONDITIONAL verdict. No change to the verdict is needed. I did not elevate secondary technical concerns (e.g., the rank assertion for the local cohomology module in Proposition 5.10) because they are likely provable; the global Selmer-rank hypothesis is the load-bearing point.","tokens_in":31245,"tokens_out":28431,"duration_ms":301259,"concrete_test":"Compute, via the control theorem at a height-one prime of Λ_gh corresponding to a classical specialization (g_y,h_z) with weights (2,1), the dimension of H^1_{G∪+}(Q, V_f ⊗ V_{g_y} ⊗ V_{h_z}(2-t_1)); if the dimension is not 1, the rank-one hypothesis used in Corollary 5.19 is false. Alternatively, derive the Λ_gh-rank by a global Euler characteristic computation and verify the parity from the epsilon factors in Assumption 3.5; a rank different from 1 would invalidate the injectivity step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 / Corollary 5.19 equates λ_{N_g}(g)·BF(f,g,h) with Ω_{f,γ}·L^p_g(f,g,h)·κ(f,g,h) by showing their images under the Perrin-Riou map L = L^{+−+}_{fgh}∘pr^{+−+}∘res_p coincide (Theorem 5.18) and then invoking injectivity of L on H^1_{G∪+}(Q,V^†_{fgh}). The injectivity is exactly the unproved hypothesis that this Selmer group is a torsion-free Λ_gh-module of rank 1 (Remark 5.20 only says it is 'expected by sign considerations' and offers H^0(Q,ρ†)=0 as a sufficient condition for torsion-freeness). If the rank is >1, the kernel of L can contain the difference of the two classes; if there is torsion, the same conclusion can fail. The same hypothesis is reused in Corollary 5.22 and Proposition 5.29, and the exceptional-zero Theorem 1.6 additionally assumes Conjecture 6.8. Thus the advertised comparison is conditional on a global Selmer-rank assertion that is not proved in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares the anticyclotomic diagonal-cycle class κ(f,g,h) with a weighted Beilinson–Flach class BF(f,g,h) attached to a triple (f,g,h) of Hida families, where f is a CM family passing through the irregular weight-one Eisenstein series Eis_1(ε_K). The main result, Theorem 1.1 / Corollary 5.19, asserts that, under a rank-one torsion-free Selmer hypothesis on H^1_{G∪+}(Q,V^†_{fgh}) and a non-vanishing hypothesis on the triple product p-adic L-function L^p_g(f,g,h), one has λ_{N_g}(g) BF(f,g,h) = Ω_{f,γ} L^p_g(f,g,h) κ(f,g,h). The proof factors the triple product p-adic L-function into two Hida–Rankin L-functions (Prop. 5.2), constructs BF satisfying the balanced local condition, and compares the images under a Perrin–Riou map L^{+−+}_{fgh} (Thm. 5.18). Injectivity of that map on the Selmer group is then invoked to identify the two global classes. A further section treats the p-exceptional case using improved L-functions, improved diagonal cycles, and a conjectural improved Beilinson–Flach class cBF, giving Theorem 1.6 / Proposition 6.18 under additional hypotheses including Conjecture 6.8.","tokens_in":31622,"tokens_out":10456,"duration_ms":124563,"significance":"If the conditional statements are accepted, the paper provides a new explicit bridge between two a priori unrelated types of Euler systems, extending the Bertolini–Darmon–Venerucci comparison of Heegner points and Beilinson–Kato classes to the setting of anticyclotomic diagonal cycles and Beilinson–Flach elements. The paper is careful and transparent about its hypotheses: Assumption 5.3, the rank-one/torsion-free Selmer assumption, and Conjecture 6.8 are all stated explicitly. The use of p-adic Artin formalism in Proposition 5.2 and the appeal to independent reciprocity laws of [KLZ17], [BSV22b], and [DR22] are coherent. The main caveat is that the central equality of global cohomology classes rests on an unproved Selmer rank hypothesis, and the exceptional-zero theorem additionally rests on an unproved conjecture; this limits the unconditional scope of the paper rather than invalidating its internal logic.","major_comments":[{"comment":"The injectivity of L^{-+}_{gh} is asserted using the fact that H^1(Q_p, V_g^-⊗V_h^+(2-t_1)) is a 'torsion-free Λ_gh-module of rank 1'. However, Lemma 5.9, which is the immediately preceding lemma, proves only torsion-freeness; no proof or reference is given for the rank-one assertion. This rank statement is load-bearing: Prop. 5.10 is used in Prop. 5.14 to show that BF(f,g,h) satisfies the balanced local condition, and it is therefore needed for the main comparison. The rank-one fact may be standard from local Euler characteristic computations, but it needs to be stated and proved or explicitly cited.","section":"§5.4, Prop. 5.10"},{"comment":"The central injectivity step in Cor. 5.19 is exactly the unproved hypothesis that H^1_{G∪+}(Q,V^†_{fgh}) is a torsion-free Λ_gh-module of rank 1. If the rank is greater than one, the kernel of L^{+−+}_{fgh}∘pr^{+−+}∘res_p can contain the difference of the two classes, and if there is torsion the same conclusion can fail. Remark 5.20 only says the rank statement is 'expected by sign considerations' and notes that H^0(Q,ρ^†)=0 would give torsion-freeness; it does not prove rank one. Since this same hypothesis is reused in Cor. 5.22 and Prop. 5.29, the paper's main theorem is conditional on a substantial Selmer-rank assertion that is not established. The authors should either prove this assertion under acceptable hypotheses or present the main theorem explicitly as a conditional result with a more thorough discussion of evidence.","section":"§5.5, Cor. 5.19 and Remark 5.20"},{"comment":"The introduction's Theorem 1.6 states the exceptional-zero comparison as an implication of the Selmer rank/torsion-freeness and non-vanishing assumptions, but it omits Conjecture 6.8. The class cBF(f,g,h) appearing in the theorem is defined in Definition 6.16 only under Assumption 6.10, which is exactly Conjecture 6.8. Thus the statement in the introduction is incomplete: without Conjecture 6.8 the object cBF need not exist. The abstract also advertises 'some arithmetic applications' without mentioning that the main results are conditional on an unproved conjecture in the exceptional-zero case. Please state all hypotheses in Theorem 1.6 and add appropriate caveats to the abstract.","section":"§6.4, Theorem 1.6 vs. Prop. 6.18"}],"minor_comments":[{"comment":"There are several spelling inconsistencies in author names: 'B¨ uy¨ ukkboduk' and 'B¨ uy¨ yukboduk' appear in the introduction and references. These should be normalized.","section":"§1, p. 1; references"},{"comment":"The statement 'h is a newform of level N_h' is ambiguous because h is a Hida family. It presumably means the specialization h_{z0} is a newform of prime-to-p conductor N_h, or that the family has tame conductor N_h. Please clarify.","section":"Assumption 5.1(iii)"},{"comment":"The proof says 'This follows as in [BSV22b, Proposition 7.3], working with V_f instead of V_f.' The last displayed object appears to be a typo; it should probably be 'V_f' or 'V_{f}' as appropriate.","section":"Prop. 5.11 proof"},{"comment":"In the sentence beginning 'As it occurs with the p-adic L-function, the factor ... appears in the interpolation property when considering its variation in families', the wording is awkward and the reference to the appearing factor is unclear. Please rephrase and make the precise factor and its role explicit.","section":"§6.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest and carefully structured, and I see no sign of circularity: the Selmer rank and non-vanishing hypotheses are explicitly declared inputs, not consequences of the conclusion. The main issue is that the central theorem is conditional on a rank-one/torsion-free Selmer statement that is not proved, and the exceptional-zero theorem additionally assumes Conjecture 6.8, which is omitted from the introduction's Theorem 1.6. The missing rank-one proof in Prop. 5.10 is likely repairable by a standard local cohomology argument, and the global Selmer rank hypothesis may be beyond the scope of the paper, but it should be framed more clearly. For a journal that welcomes conditional theorems with explicitly stated hypotheses, a revision addressing these points could be suitable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper does something real. It extends the Bertolini-Darmon-Venerucci comparison to anticyclotomic diagonal cycles and cyclotomic Beilinson-Flach elements, working through a CM Hida family that degenerates to an irregular weight-one Eisenstein series. The new construction is the weighted Beilinson-Flach class BF(f,g,h), and the proof of the local comparison in Theorem 5.18 is detailed and coherent. The factorization of the triple product p-adic L-function as a product of two Hida-Rankin p-adic L-functions (Prop 5.2) follows from Artin formalism and is clean.\n\nWhat I like: the paper is transparent about its assumptions. Theorem 1.1 is explicitly conditional on two hypotheses: the Selmer group H^1_{G∪+}(Q,V^†_{fgh}) being a torsion-free Λ_gh-module of rank 1, and the nonvanishing of L^p_g(f,g,h). The first is the load-bearing one. Corollary 5.19 identifies the two classes by showing their images under a Perrin-Riou map coincide and then invoking injectivity of that map, which is exactly the unproved rank-one/torsion-free statement. Remark 5.20 says it is expected by sign considerations, with a sufficient torsion-freeness condition. That is honest, but it means the advertised comparison is not unconditional.\n\nThe exceptional-zero section is more speculative: Theorem 1.6 depends on Conjecture 6.8, the existence of an improved Beilinson-Flach class. The authors present this as a standard expectation. Fine, but it lowers confidence in the arithmetic applications.\n\nI don't see circularity. The reciprocity laws come from KLZ17 and BSV22b/DR22, and the p-adic L-function factorization is derived, not assumed. The paper is well-written and the references are appropriate.\n\nWho is this for? People working in Iwasawa theory, p-adic L-functions, and Euler systems. A serious referee should verify the local Perrin-Riou computations in Section 5.4 and consider whether the rank-one/torsion-free assumption can be replaced by a more accessible hypothesis. The paper deserves refereeing, not a desk reject. I'd take the main theorem as a conditional result likely true under the stated hypotheses, but the arithmetic applications should not be quoted without the caveats.","headline":"A well-built, transparent conditional comparison of diagonal cycles to Beilinson-Flach elements; the advertised equality rests on an unproved rank-one Selmer hypothesis, and the exceptional-zero applications on a conjecture.","tokens_in":32065,"tokens_out":2537,"would_cite":true,"duration_ms":29137,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R23","11F85"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, under a rank-one torsion-free Selmer assumption, the weighted Beilinson–Flach class BF(f,g,h) lies in the balanced Selmer group and equals Ω_{f,γ} L^p_g(f,g,h) times the anticyclotomic diagonal-cycle class κ(f,g,h),","keywords":["anticyclotomic Euler systems","Beilinson–Flach classes","diagonal cycles","Hida families","p-adic L-functions","Perrin-Riou maps","CM degeneration","irregular weight-one Eisenstein series"],"falsifier":"Specialize the equality at a good crystalline point (y,z) with L^p_g(f,g,h)≠0 and compute the Perrin-Riou images of both classes using the interpolation formulas of Theorems 4.8 and 4.15: if the ratio of the images is not Ω_{f,γ}^{-1}λ_{N_g}(g)L^p_g(f,g,h), Theorem 5.18 fails. A direct computation of H^1_{G∪+}(Q,V^†_{fgh}) at one arithmetic point would also settle the rank-one torsion-free premise.","tokens_in":31115,"feed_emoji":"🔗","tokens_out":12980,"duration_ms":123217,"temperature":0.7,"pith_summary":"This paper claims that two very different p-adic cohomology classes attached to the triple of Hida families (f,g,h)—the anticyclotomic diagonal cycle class and a suitably weighted Beilinson–Flach class—are actually the same class up to an explicit scalar. The scalar is the product of a p-adic period Ω_{f,γ} and the g-unbalanced triple product p-adic L-function L^p_g(f,g,h), and the equality takes place in the balanced Selmer group once a rank-one torsion-free Selmer assumption is granted. The weight-one input f is the irregular Eisenstein series attached to an imaginary quadratic field K, so its Galois representation splits into the cyclotomic and quadratic characters L(1)⊕L(ε_K), and the Beilinson–Flach system correspondingly splits into two Rankin–Selberg pieces. The proof's engine is a factorization of L^p_g(f,g,h)^2 as a product of two Hida–Rankin p-adic L-functions, followed by an injectivity argument through Perrin-Riou maps. If correct, this is the anticyclotomic analogue of the earlier Heegner-point/Beilinson–Kato comparison.","feed_headline":"A p-adic equality links diagonal cycles and Beilinson–Flach classes","feed_subtitle":"Under a rank-one Selmer assumption, the two classes coincide up to a p-adic period and the triple-product L-value.","key_machinery":"The central object is the weighted Beilinson–Flach class BF(f,g,h)=v_{f,1}⊗L_p(g,h⊗ε_K)κ_{g,h}+v_{f,ε_K}⊗L_p(g,h)κ_{g,h⊗ε_K}, built from the two Beilinson–Flach classes κ_{g,h} and κ_{g,h⊗ε_K} after the CM family f degenerates to the irregular weight-one Eisenstein series. The identity that carries the argument is the factorization of the triple product p-adic L-function into a product of two Hida–Rankin p-adic L-functions (Proposition 5.2), which is a p-adic Artin-formalism consequence of the decomposition V_f ≅ L(1)⊕L(ε_K). The injectivity of the Perrin-Riou map L^{-+}_{gh} (Proposition 5.10) then forces the two classes to be proportional, and the proportionality constant is read off from","core_discovery":"Under Assumption 5.3 (the triple product p-adic L-function L^p_g(f,g,h) is not identically zero) and the unproved structural assumption that H^1_{G∪+}(Q,V^†_{fgh}) is a torsion-free Λ_{gh}-module of rank one, the paper proves that the weighted Beilinson–Flach class BF(f,g,h) lies in the balanced Selmer group and satisfies λ_{N_g}(g)·BF(f,g,h)=Ω_{f,γ}·L^p_g(f,g,h)·κ(f,g,h). Here BF(f,g,h) is assembled from the Beilinson–Flach classes of (g,h) and (g,h⊗ε_K), each weighted by the corresponding Hida–Rankin p-adic L-function, after specialization of the CM family f to the irregular Eisenstein series f=Eis_1(ε_K); κ(f,g,h) is the balanced diagonal-cycle class. The identification is made by compari","pith_inferences":["If the rank-one torsion-free Selmer assumption can be proved by control theorems—the authors note it is expected from sign considerations—the same argument should give the equality unconditionally for this family of triples, without changing the reciprocity-law core.","The mechanism should extend to any CM Hida family specializing to a weight-one Eisenstein series, not only Eis_1(ε_K); the paper's Remark 2.3 indicates the broader class, so the same comparison should hold for those twist families.","A natural construction to attempt next is the improved Beilinson–Flach class bκ_{g,h} directly from Rankin–Eisenstein classes, which would upgrade the exceptional-zero theorem from conditional to unconditional.","Because the diagonal-cycle construction does not use modular units, the same comparison may be reproducible in settings where Beilinson–Flach classes are not available, such as Shimura curves, giving anticyclotomic comparisons entirely through geometric classes."],"forward_implications":["For fixed good crystalline specializations g=g_{y0}, h=h_{z0}, the equality descends to a relation between the specialized weighted Beilinson–Flach class and the specialized diagonal cycle class, whenever the balanced Selmer group is one-dimensional and L^p_g(f,g_α,h_α)≠0 (Corollary 5.25).","The Perrin-Riou image of BF under Log_{ω_g⊗ω_h} factors as Ω_{f,γ} L^p_g(f,g,h) L^p_f(f,g,h)/λ_{N_g}(g), giving a purely p-adic factorization of a big logarithm by two triple-product L-functions (Corollary 1.4/5.29).","In the exceptional-zero case, the improved classes satisfy Ω_{f,γ} cL^p_g(f,g,h) bκ(f,g,h)=λ_{N_g}(g) cBF(f,g,h) (Theorem 1.6/6.17-6.20), so the comparison survives the vanishing of the relevant Euler factors once the predicted improved Beilinson–Flach class is available.","The result supplies a new instance of the principle that cyclotomic and anticyclotomic Euler systems can be matched after a degeneration: diagonal cycles here play the role of Heegner points, Beilinson–Flach classes the role of Kato classes.","A byproduct is a mixed factorization (Corollary 5.22) relating L^p_g and L^p_h to Hida–Rankin L-functions with and without the quadratic twist ε_K, even though the interpolation regions are disjoint."],"supporting_citations":[{"why":"Model for comparing a geometric Euler system with a cyclotomic one; its Heegner-point/Beilinson–Kato identity is what this paper extends.","marker":"[BDV22]"},{"why":"Constructs the diagonal-cycle classes and proves the explicit reciprocity laws used for κ(f,g,h).","marker":"[BSV22b]"},{"why":"Provides p-adic families of diagonal cycles, an alternate source of the diagonal-cycle class.","marker":"[DR22]"},{"why":"Constructs Beilinson–Flach classes and supplies the Hida–Rankin L-functions, Perrin-Riou maps, and reciprocity laws.","marker":"[KLZ17]"},{"why":"Establishes the interpolation formula for the triple product p-adic L-function L^p_g(f,g,h).","marker":"[Hsi21]"},{"why":"Proves f_α is an étale weight-one Eisenstein point and gives the CM Hida family f.","marker":"[BDP22]"},{"why":"Provides the isomorphism V_f ≅ Ind^Q_K Λ_f(φ), the structural input behind the decomposition into cyclotomic and quadratic characters.","marker":"[BSTW24]"},{"why":"Supplies the Eisenstein-degeneration framework for comparing Euler systems, against which this paper's CM degeneration is positioned.","marker":"[LR24]"},{"why":"Gives the anticyclotomic Euler-system interpretation of diagonal cycles and the Selmer argument used in Corollary 5.25.","marker":"[ACR23b]"}],"fun_headline_variants":["Diagonal cycles equal Beilinson–Flach up to a p-adic period","CM Eisenstein bridge: from Heegner to Beilinson–Flach","p-adic L-functions tie two Euler systems together","Anticyclotomic and cyclotomic classes coincide under Selmer rank one","Eisenstein degeneration bridges two Euler systems"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The comparison rests on an unproved structural premise: the relevant Selmer group is a torsion-free module of rank one over the Iwasawa algebra of the two Hida families, which the authors expect from sign considerations but do not prove; if that fails, the injectivity step identifying the two classes collapses, and in the exceptional-zero section the existence of an improved Beilinson–Flach class is also assumed.","fun_headline_variants_meta":{"raw":{"variants":["Diagonal cycles equal Beilinson–Flach up to a p-adic period","CM Eisenstein bridge: from Heegner to Beilinson–Flach","p-adic L-functions tie two Euler systems together","Anticyclotomic and cyclotomic classes coincide under Selmer rank one","Eisenstein degeneration bridges two Euler systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000697,"raw_usage":{"total_tokens":2977,"prompt_tokens":725,"completion_tokens":2252,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":469,"completion_tokens_details":{"reasoning_tokens":2159}},"tokens_in":469,"tokens_out":2252,"duration_ms":19168,"temperature":1.0,"reasoning_tokens":2159,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T21:56:24.813337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Specialize the equality at a good crystalline point (y,z) with L^p_g(f,g,h)≠0 and compute the Perrin-Riou images of both classes using the interpolation formulas of Theorems 4.8 and 4.15: if the ratio of the images is not Ω_{f,γ}^{-1}λ_{N_g}(g)L^p_g(f,g,h), Theorem 5.18 fails. A direct computation of H^1_{G∪+}(Q,V^†_{fgh}) at one arithmetic point would also settle the rank-one torsion-free premise.","supporting_citations":[],"review_version":1}