{"id":"2031db20-27a0-426e-bfd7-240045ffb196","arxiv_id":"2501.15336","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs a new anticyclotomic Euler system for Asai Galois representations attached to p-ordinary Hilbert modular forms, with norm relations and p-adic interpolation.","lead":"This paper constructs a new family of cohomology classes, called an anticyclotomic Euler system, for four-dimensional Galois representations attached to Hilbert modular forms over real quadratic fields. If the classes are nonzero, they imply the Bloch-Kato conjecture in rank one and a divisibility in the Iwasawa main conjecture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.7's norm relations rest on an unverified 'one can check' congruence and unchecked application of Rubin's Lemma 9.6.1; Proposition 5.6's displayed formula appears to duplicate terms, so the underlying computation needs independent verification.","rationale":"The reader's weakest assumption is precisely the load-bearing point. Theorem 5.7 is the central claim, and its proof contains a single sentence 'One can check' for the congruence that is then converted to an exact equality by Rubin's lemma. The proof neither displays the computation nor verifies the integrality and local hypotheses of Rubin's lemma. If this step fails, there is no Euler system, and the applications in Section 7 collapse. Our independent reading found an additional red flag: the displayed formula in Proposition 5.6, which is the closest explicit norm-relation statement for the big class, appears to repeat the first two terms at the end, indicating that the 'tedious computation' may contain an error. This does not make us reject the paper, because the gap is plausibly repairable and the surrounding framework (geometric cycles, distribution relations) is standard and carefully developed; it does support the reader's CONDITIONAL verdict. We therefore see no reason to change the verdict.","tokens_in":28944,"tokens_out":10869,"duration_ms":92699,"concrete_test":"Directly compute the difference cor_{K[nq]/K[n]} κ*_{∞,nq}(F,G) - P_q(V_{G,ψ};Fr_q^{-1}) κ*_{∞,n}(F,G) using the three relations in Proposition 5.5 and the definition of κ* (bottom of p.23), for a small split prime q; check that it vanishes modulo (q-1) in H^1_Iw(K[np∞], T) and that κ* is integral, which is the condition for Rubin's Lemma 9.6.1. Also inspect whether Proposition 5.6's duplicated terms are a typo.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central Euler system norm relations in Theorem 5.7 are obtained as follows: after defining modified classes κ*_{∞,n}(F,G) (p.23), the proof asserts 'One can check' that cor_{K[nq]/K[n]} κ*_{∞,nq}(F,G) ≡ P_q(V_{G,ψ(κ^{k-2}_{ac})}; Fr_q^{-1}) κ*_{∞,n}(F,G) (mod q-1), then invokes [Rub00, Lem. 9.6.1] to replace the congruence by an equality, and [Rub00, Thm. 6.3.5] to remove the twist. Neither the congruence nor the hypotheses of Rubin's lemma are verified: the modified classes are not shown to lie in the integral Iwasawa cohomology H^1_Iw(K[np∞], T) for a fixed lattice T, and the local conditions required by the lemma are not checked. If the congruence is off by a factor that is not 1 modulo (q-1), or if integrality fails, the exact norm relations do not follow and the Euler system may not exist. A concrete warning sign is Proposition 5.6: its displayed formula appears to repeat the first two terms after '-q^{-1}(q^2+1)' (the terms with -η1(q)... and +qω1(q)...), suggesting a possible copying or algebra error in the computation on which the congruence is based. This makes the delegated 'one can check' step the most load-bearing unverified point in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs an anticyclotomic Euler system for the Asai Galois representation attached to a p-ordinary Hilbert modular form g over a real quadratic field F, twisted by a CM Hecke character ψ. The construction uses generalized Hirzebruch–Zagier cycles obtained from modular curves of varying level diagonally embedded into products with Hilbert modular surfaces, following the approach of Darmon–Rotger. The main theorems are: (A) the existence of an anticyclotomic Euler system {κ_{ψ,g,n,∞}} with the expected norm relations, (B) applications to the Bloch–Kato conjecture in rank one, and (C) applications to the Iwasawa main conjecture, conditional on big-image hypotheses and on the framework of the unpublished Jetchev–Nekovář–Skinner manuscript. The paper also proves p-adic interpolation of the classes along Hida families.","tokens_in":29260,"tokens_out":5019,"duration_ms":42823,"significance":"If the main construction is correct, this is an important new Euler system: it is the first anticyclotomic Euler system for Asai representations over real quadratic fields, and the p-adic interpolation result is a substantial new ingredient. The geometric lemmas in Sections 4 and 5.2 (Lemmas 4.1–4.4, 5.3, 5.4) are proven in detail, and the construction of the big cohomology classes is clearly presented. The applications, while conditional on the unpublished work [JNS24], are natural and would be significant. The paper is well written and carefully signposted. However, the central norm-relation step in Theorem 5.7 contains a deferred 'one can check' congruence whose hypotheses and proof are not supplied, and Proposition 5.6 shows evidence of a transcription error. These issues are load-bearing for the main theorem.","major_comments":[{"comment":"The displayed formula for cor_{K[nq]/K[n]}κ_{∞,n}(F,G) contains a likely duplication: after the term '-q^{-1}(q^2+1)' the expression repeats the first two summands exactly (the terms with -η1(q)κ_1^{-1/2}(q)a_{q1}(G)a_{q2}(G)(...) and +qω1(q)χ_G(̟_{q1})χ_G(̟_{q2})(...)^2). This suggests a copying error in the computation, which is delegated to 'somewhat tedious computation'. Since this formula is the basis for the congruence used in Theorem 5.7, the formula must be corrected and the computation either supplied in detail or independently verified.","section":"§5.2, Proposition 5.6"},{"comment":"The proof asserts 'One can check' the congruence cor_{K[nq]/K[n]}κ*_{∞,nq}(F,G) ≡ P_q(VG,ψ(κ^{k-2}_{ac}); Fr_q^{-1})κ*_{∞,n}(F,G) mod (q-1), and then invokes [Rub00, Lem. 9.6.1] to replace this congruence by an equality. However, the hypotheses of Rubin's lemma are not verified: the classes κ*_{∞,n}(F,G) are not shown to lie in the integral Iwasawa cohomology H^1_Iw(K[np^∞], T) for a fixed O-lattice T, and the required local conditions are not checked. Without these checks, the exact norm relations of Theorem 5.7 do not follow.","section":"§5.2, proof of Theorem 5.7"},{"comment":"The application of [Rub00, Thm. 6.3.5] to remove the twist by κ^{k-2}_{ac} is not justified. The character κ^{k-2}_{ac} is an infinite-order anticyclotomic character with values in Λ^×, and the hypotheses of the twisting theorem (which typically requires finite-order twists or specific integrality conditions) are not discussed in the proof. This is a load-bearing step in obtaining the untwisted norm relations with P_q(VG,ψ; Fr_q^{-1}).","section":"§5.2, proof of Theorem 5.7"},{"comment":"The transition from Proposition 5.5 to Theorem 5.7 is not fully demonstrated. In particular, the definition of the modified classes κ*_{∞,n}(F,G) involves multiplication by a product over q|n of terms depending on ψ_P(Fr_q) and Fr_q, and it is not shown that these operations preserve the integral classes required for Rubin's lemma. The congruence modulo (q-1) is stated without a derivation from the (currently defective) Proposition 5.6, so the proof of Theorem 5.7 is incomplete at this point.","section":"§5.2, Proposition 5.5 to Theorem 5.7"}],"minor_comments":[{"comment":"The abstract contains the typo 'emdedded' for 'embedded'.","section":"Abstract"},{"comment":"In the statement of Theorem A, the collection is written as {κψ,g,n,∞ : m∈S}; the dummy variable should be n, not m.","section":"Introduction, Theorem A"},{"comment":"In the definition of κ_{∞,n}(F,G), the notation [ξ_{̟_{n1}}^{-1}] is used without prior definition; the notation for the diamond operator on the Hilbert modular form side should be clarified.","section":"§5.2, equation (3) and following"},{"comment":"In the third displayed relation of Lemma 5.4, the term '(q+1)(⟨1,⟨̟_{q1}⟩)' should presumably be '(q+1)(1,⟨̟_{q1}⟩)'; the ⟨1 is likely a typo.","section":"§5.2, Lemma 5.4"},{"comment":"The reference [NN16] is corrupted: 'Wies/suppress lawa Nizio/suppress l' should read 'Wiesława Nizioł'. The reference [ACR23b] is listed as 'to appear' but is used for several technical results; if it is not yet published, the dependence should be clearly flagged in the text.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's main applications (Theorems B and C) rely on the unpublished preprint [JNS24] and on results from the authors' previous work [ACR23b]. This is a serious dependence, and the editor may wish to consider whether the results should be stated conditional on the acceptance of those works. More importantly, the norm-relation proof in Theorem 5.7 is not complete as written: the 'one can check' step and the application of Rubin's lemma require verifiable hypotheses, and Proposition 5.6 contains a possible duplication that suggests the underlying computation needs independent checking. These are fixable within the scope of the manuscript, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take on Alonso–Castella–Rivero, arXiv:2501.15336. The genuinely new thing is an anticyclotomic Euler system for the Asai representation of a p-ordinary Hilbert modular form over a real quadratic field, interpolated in Hida families. That is not a routine extension: the earlier Asai–Flach system of Lei–Loeffler–Zerbes is cyclotomic, and the Hirzebruch–Zagier cycles here are made to vary with tame level and then pushed through the JNS anticyclotomic machinery. If the norm relations hold, Theorems B and C are meaningful rank-one Bloch–Kato and Iwasawa main conjecture results.\n\nWhat is good: the geometric part is mostly real work. Lemmas 4.3, 5.3 and 5.4 give the cycle and class distribution relations with proofs, not citations. The dependence on the authors’ earlier ACR23b is for technical lemmas and Selmer comparisons, not for the theorem being proved. I saw no circularity and no fitted parameters.\n\nWhere it gets soft: Theorem 5.7, the central existence theorem, is completed by a “one can check” congruence modulo (q−1), followed by an invocation of Rubin’s Lemma 9.6.1 and Theorem 6.3.5 without verifying the hypotheses. The modified classes κ* are not shown to lie in the integral Iwasawa cohomology for a fixed lattice, and the local conditions Rubin needs are not checked. This matters because Proposition 5.6, which is supposed to supply the exact norm relation, contains a displayed formula that literally repeats the first two terms after the constant −q^{-1}(q^2+1). That looks like a copying or sign error in the “somewhat tedious computation.” A referee needs to redo that computation from Lemma 5.2 and Proposition 5.5 independently. The applications also rely on the unpublished Jetchev–Nekovář–Skinner preprint; that is a dependency, not a flaw, but it makes the theorem’s claims conditional.\n\nBottom line: the construction is plausible and the geometric framework is solid enough that the paper deserves refereeing, but I would not take Theorem 5.7 on faith. Send it to a referee who can verify the algebra.","headline":"A serious and plausible construction of an anticyclotomic Euler system for Asai representations, but the central norm relations are delegated to an unverified congruence and a suspicious duplicated formula in Proposition 5.6; needs a careful referee before the applications can be trusted.","tokens_in":29804,"tokens_out":3477,"would_cite":true,"duration_ms":30626,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F41","11F80","11G18","11R23"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs an anticyclotomic Euler system for Asai Galois representations attached to p-ordinary Hilbert modular forms over real quadratic fields, with classes varying in p-adic Hida families.","keywords":["Euler systems","Hirzebruch–Zagier cycles","Asai representations","Hilbert modular forms","p-adic Hida families","anticyclotomic Iwasawa theory","Bloch–Kato conjecture","Selmer groups"],"falsifier":"Compute explicitly the modified classes κ^*_{∞,n}(F,G) attached to a concrete Hilbert modular form g and Hecke character ψ at a prime q that splits in both fields. If the corestriction congruence modulo (q−1) is not an equality (or if one of the classes fails integrality at q), then the lemma cannot be applied and the claimed Euler system norm relations of Theorem 5.7 fail.","tokens_in":28746,"feed_emoji":"🌀","tokens_out":5907,"duration_ms":48283,"temperature":0.7,"pith_summary":"This paper constructs an anticyclotomic Euler system for the four-dimensional Asai Galois representation attached to a p-ordinary Hilbert modular form over a real quadratic field. The classes are built from Hirzebruch–Zagier cycles, so they come from algebraic cycles, and they are shown to interpolate in p-adic Hida families. If the construction works, Kolyvagin's method turns the non-vanishing of one class into a rank-one bound on the Bloch–Kato Selmer group and a divisibility in the anticyclotomic Iwasawa main conjecture, giving new evidence for two central conjectures in the arithmetic of automorphic forms.","feed_headline":"Hirzebruch–Zagier cycles yield a new anticyclotomic Euler system","feed_subtitle":"Classes vary in p-adic Hida families and imply rank-one Bloch–Kato and Iwasawa main conjecture results.","key_machinery":"The machinery is the system of generalized Hirzebruch–Zagier cycles: for each p-power level α, a modular curve is embedded diagonally into the product Y_1(p^α)×S_1(p^α) of a modular curve and a Hilbert modular surface, producing codimension-two cycles Δ_α[t0,t1] over the cyclotomic extension Q(ζ_{p^α}). Their étale Abel–Jacobi images, after applying Hecke idempotents and Atkin–Lehner maps, yield big cohomology classes κ_∞(F,G) valued in the tensor product of the big Galois representation of an elliptic Hida family and the big Asai representation of a Hilbert Hida family. Tame-level variants at n give classes whose corestrictions are controlled by degeneracy maps; the Euler-system norm relations are then derived from explicit relations among the cycles under these maps, with a standard lemma for Euler systems converting congruences modulo (q−1) into exact equalities.","core_discovery":"The central claim is Theorem A (Theorem 5.7): under the hypotheses that p splits in K, p does not divide the class number of K, and g is p-ordinary, there exists a collection of classes κ_{ψ,g,n,∞} in the balanced Selmer group $Sel^{{bal}}$(K[np^∞],T) indexed by squarefree n whose prime factors split in both F and K, satisfying the Euler system norm relation cor_{K[nq]/K[n]}(κ_{ψ,g,nq,∞}) = P_q(V;$Fr_q^{{-1}}$)κ_{ψ,g,n,∞} for every such prime q. The collection is an anticyclotomic Euler system for the conjugate self-dual representation V = As(V_g)|_{G_K}($ψ_P^{{-1}}$)(2-l-k/2). The proof passes through two-variable big cohomology classes κ_∞(F,G) attached to a CM Hida family F and a parallel-weight Hida family G, specializing to the desired classes, and the norm relations are obtained from explicit relations among the underlying algebraic cycles under degeneracy maps.","pith_inferences":["An explicit reciprocity law relating the big classes to the p-adic Asai L-function would turn the divisibility of Theorem 7.2 into an equality, a natural extension the authors plan for a sequel.","In the degenerate case where g is the base-change of an elliptic modular form, the Asai representation contains Sym^2 of the elliptic form as a summand; the Euler system may then be decomposed to attack the anticyclotomic Iwasawa main conjecture for the adjoint representation.","The congruence-to-equality step via the standard Euler-system lemma is the part most sensitive to integrality; an explicit computation of the modified classes at a single split prime would either confirm the norm relations or expose a missing hypothesis."],"forward_implications":["If Theorem A holds, the non-vanishing of the class κ_{ψ,g} forces the balanced Bloch–Kato Selmer group Sel^{bal}(K,V) to be one-dimensional over E in the range k<2l (Theorem 7.1).","A non-torsion class κ_{ψ,g,∞} in the Iwasawa cohomology implies that both the balanced Selmer group and its Pontryagin dual have Λ_F-rank one, with a divisibility of characteristic ideals (Theorem 7.2).","The Euler system classes and their norm relations vary in p-adic Hida families, so the construction is compatible with specialization at arithmetic points.","The construction yields an anticyclotomic analogue of the Asai–Flach Euler system, where the base field is a real quadratic field and the tower is anticyclotomic over an imaginary quadratic field."],"supporting_citations":[{"why":"Supplies the anticyclotomic Euler system formalism, balanced Selmer groups, and rank-one bounds used in Theorems B and C.","marker":"[JNS24]"},{"why":"Provides the lemma that converts norm congruences into exact norm relations, a key step in the proof of Theorem 5.7.","marker":"[Rub00]"},{"why":"Gives the template for constructing big cohomology classes from diagonal cycles in p-adic families.","marker":"[DR22]"},{"why":"Constructs the Asai–Flach Euler system, the analogue of which is the subject of this paper, and gives the Asai representation filtrations.","marker":"[LLZ18]"},{"why":"Supplies the norm-relation arguments, Selmer-group results, and hypotheses (HS)/(HW) adapted to the present setting.","marker":"[ACR23b]"},{"why":"Provides the theorem identifying the balanced Selmer group, used to place the constructed classes in Sel^{bal}.","marker":"[NN16]"},{"why":"Previous study of generalized Hirzebruch–Zagier cycles and their Abel–Jacobi images, source of many geometric lemmas used here.","marker":"[FJ24]"},{"why":"Influences the use of Kolyvagin methods with level-raising congruences in the anticyclotomic setting.","marker":"[BD05]"}],"fun_headline_variants":["Hirzebruch–Zagier cycles forge anticyclotomic Euler system","Anticyclotomic Euler system from Hirzebruch–Zagier cycles","p-adic Hida families of Euler classes from Hirzebruch–Zagier cycles","New anticyclotomic Euler system via Hirzebruch–Zagier cycles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is a lemma in the Euler-system formalism that upgrades a norm congruence modulo (q−1) to an exact equality; it applies only if the modified classes are integral and meet the required local conditions, and the proof asserts these hypotheses without a detailed check.","fun_headline_variants_meta":{"raw":{"variants":["Hirzebruch–Zagier cycles forge anticyclotomic Euler system","Anticyclotomic Euler system from Hirzebruch–Zagier cycles","p-adic Hida families of Euler classes from Hirzebruch–Zagier cycles","New anticyclotomic Euler system via Hirzebruch–Zagier cycles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001149,"raw_usage":{"total_tokens":4753,"prompt_tokens":920,"completion_tokens":3833,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":3738}},"tokens_in":536,"tokens_out":3833,"duration_ms":24521,"temperature":1.0,"reasoning_tokens":3738,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:23:36.437246+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute explicitly the modified classes κ^*_{∞,n}(F,G) attached to a concrete Hilbert modular form g and Hecke character ψ at a prime q that splits in both fields. If the corestriction congruence modulo (q−1) is not an equality (or if one of the classes fails integrality at q), then the lemma cannot be applied and the claimed Euler system norm relations of Theorem 5.7 fail.","supporting_citations":[],"review_version":1}