{"id":"6b74b419-61d1-4439-ad6a-51ed624116a1","arxiv_id":"2506.19673","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Asai-Flach Euler system of Lei-Loeffler-Zerbes is interpolated p-adically over Hida families of Hilbert modular forms over real quadratic fields.","lead":"This paper shows that the Asai-Flach Euler system for Hilbert modular forms over a real quadratic field varies p-adically in Hida families. The result supplies a key ingredient in a recent proof of the Bloch-Kato conjecture in analytic rank zero for Asai representations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Split-prime p-stabilization relation in Theorem 4.1.3 rests on a delegated, normalization-sensitive construction; if the proportionality constant is off by a unit, the explicit interpolation factors in Theorem C are wrong.","rationale":"I read the paper's main claim as the construction of a Hida-family class c_AF(Pi) whose specializations agree, with explicit scalar factors, with the Asai-Flach classes of [LLZ18]. The existence of a family Galois representation and of a family class follows from the control theorem (Theorem 3.2.1) and the pushforward of the Iwasawa class; those parts are standard extensions of [She25] and [LLZ18], and I found no circularity or fitted parameters. The genuinely fragile step is the split-prime p-stabilization relation, because it is the only place where the explicit interpolation factor is determined, and because the paper's proof is not self-contained: Proposition 6.2.1 is asserted with a reference rather than a construction. The reader's weakest_assumption identifies exactly this point, and I agree. I do not think the paper should be rejected: the theorem is plausible, the cited [Gro20] is published, and the inert-prime case is proved directly, with supporting evidence in the form of explicit norm relations. But the split-prime normalization should be verified before Theorem C's explicit factors are relied upon, so the CONDITIONAL verdict is appropriate and no verdict change is needed from my pass.","tokens_in":14282,"tokens_out":21230,"duration_ms":218998,"concrete_test":"Check the split-prime case of Theorem 4.1.3 in the special situation where the primes above p are trivial in the narrow class group, which Section 6.2 states admits a direct proof similar to the inert case. Recompute the p-stabilization relation from [LLZ18, Cor. 7.4.2] and the degeneracy maps Pr_{p,1}, Pr_{p,2} exactly as in Section 6.1, and compare the resulting R(X) with the three-factor polynomial in Theorem 4.1.3. If the direct computation gives a different polynomial or a different normalization, the proportionality constant in Proposition 6.2.1 is off and Theorem C's interpolation factors need correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central output is the explicit specialization formula of Theorem C (Theorem 5.4.3). Its passage from the p-stabilized class AF[Pi,alpha,j] to the newform-level class AF[Pi,j] uses Theorem 4.1.3, whose split-prime case is proved in Section 6.2. The proof of the existence and normalization of the 'motivic' functional Z_mot,j in Proposition 6.2.1 is not actually given: the proof says it 'amounts to nothing more than carefully keeping track of all the choices' and refers to [LZ24] and [Gro20]. The subsequent argument identifies Z_mot,j with Z_an,j tensor z_0 using Lemma 6.2.2, and fixes z_0 = AF[Pi,j] by the spherical normalization Z_an(W_sph, ch(Z_p^2)) = 1. If either the construction of Z_mot,j fails to satisfy the second equality in Proposition 6.2.1, or the analytic spherical normalization differs by a unit, then the proportionality in Theorem 4.1.3 holds only up to a unit, and the scalar factors in Theorem C are off by that unit. Since Theorem C's interpolation factors are the only stated compatibility with the [LLZ18] classes, this would break the advertised interpolation even if the family class c_AF(Pi) itself exists.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper concerns the p-adic variation of the Asai–Flach Euler system for Hilbert modular forms over a real quadratic field. The authors define an Iwasawa cohomology class z∞ in the ordinary part of the inverse-limit cohomology of a Hilbert modular variety tower, and then, after fixing an ordinary Hida family Π, they construct a class c_AF(Π) valued in a rank-4 family of Galois representations M(Π)^*. Their main result, Theorem 5.4.3, asserts that at algebraic weight characters (λ,h) this class specializes to an explicit scalar multiple of the classical Asai–Flach class AF[Π,j] of Lei–Loeffler–Zerbes. The proof scheme is: prove/adapt control theorems from [She25] to produce the ordinary cohomology module; construct z∞ via pushforward from a G* Shimura variety; prove a compatibility of moment maps; and finally relate the p-stabilized and newform-level Asai–Flach classes via a p-stabilization relation (Theorem 4.1.3). The last step, for split primes, is the most delicate and is treated in Section 6.","tokens_in":14548,"tokens_out":9358,"duration_ms":85393,"significance":"If the theorems are correct, this is a valuable and timely contribution: it gives a p-adic interpolation of an Euler system in the Hilbert modular setting with explicit interpolation factors, and it is already used as input in Grossi–Loeffler–Zerbes' recent work on the Bloch–Kato conjecture for Asai representations. The main theorems are stated with precision, and Theorem 5.4.3 in particular is a concrete, checkable identity. The paper is a natural sequel to [She25] and [LLZ18] and is clearly organized. The main difficulty is that two load-bearing ingredients—the generalized control theorem and the split-prime p-stabilization relation—are not fully proved here but are delegated to prior work, with the split-prime part resting on normalization-sensitive constructions that are not pinned down in the text.","major_comments":[{"comment":"The split-prime case of the p-stabilization relation is load-bearing for Theorem 5.4.3, because the factor R(p^{-1-h}) in Theorem C comes from Theorem 4.1.3. However, the existence and normalization of the motivic functional Z_mot,j is not proved in the manuscript; the proof says that the construction 'amounts to nothing more than carefully keeping track of all the choices' and refers to [LZ24] and [Gro20]. A reader cannot verify from the present text that the second equality in Proposition 6.2.1 holds with the factor (p^2-1)^{-1} or that the spherical normalization Z_mot,j(W_sph, ch(Z_p^2)) = AF[Π,j] is free of a unit. Since Lemma 6.2.2 only shows that the space of such functionals is at most one-dimensional, any undetected unit error in either normalization would change the scalar in Theorem 5.4.3 by that unit and break the advertised compatibility with the [LLZ18] classes. I request that the proof of Proposition 6.2.1 be included, or at minimum that the precise statements from [LZ24] and [Gro20] that imply it be quoted and their hypotheses checked.","section":"§6.2, Proposition 6.2.1 and Theorem 4.1.3"},{"comment":"The proof of Lemma 6.2.2 asserts without proof that Z_an,j(W_sph, ch(Z_p^2)) = 1 and that the Rankin–Selberg integral is non-zero on spherical data. The value of this integral is normalization-sensitive: it depends on the choice of Haar measure on N(Q_p)\\GL_2(Q_p), the additive character defining the Whittaker model, and the Godement–Siegel section f_Φ. Since the final formula R(p^{-1-h}) is obtained by comparing this value with the computed value on the p-stabilized vector W_α and ch((0,1)+pZ_p^2), a different normalization convention would alter the displayed scalar in Theorem 5.4.3. The manuscript should either give the computation or cite a precise statement in the literature that fixes the same conventions as [LLZ18].","section":"§6.2, Lemma 6.2.2"},{"comment":"Theorem 3.2.1 is the foundation for the construction of M(Π)^* and for the specialization step in Theorem 5.4.3, but its proof is delegated: 'This is proved in exactly the same way as Corollary 3.14 of [She25]'. The new feature here is that the coefficient sheaves have non-trivial central character factoring through the norm map, and the subgroups K_{n,p} are defined using E_K(p). Since the moment-map compatibility (†) and the Tor spectral sequence are asserted for this more general setting, the paper should state which parts of [She25] carry over unchanged and where the central-character condition is used. As written, a reader cannot check that the spectral sequence has the claimed abutment.","section":"§3.2, Theorem 3.2.1 and §3.3, Definition 3.3.1"}],"minor_comments":[{"comment":"The symbol k' in \\binom{k}{j}\\binom{k'}{j} is not defined; presumably k and k' are k_1 and k_2, but this should be stated explicitly.","section":"§5.4, Theorem 5.4.2"},{"comment":"The phrase 'As before write and h=t_1+t_2+j' appears to have a missing word before 'and'; probably 'write h=t_1+t_2+j'.","section":"§5.4, Theorem 5.4.2"},{"comment":"The notation H^d_et(Y_G(K∞)_Q,O) is used for cohomology of the inverse-limit tower, while §3.1 defines H^i_Iw for this object. The notation should be harmonized or explained.","section":"§3.3, Definition 3.3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is heavily dependent on the authors' own prior and concurrent work. This is not in itself a problem, but the editor may want to verify that the statements from [LZ24] and [Gro20] do establish exactly the normalization used in Proposition 6.2.1. The split-prime p-stabilization relation is the only part of the proof that is both essential for Theorem 5.4.3 and not demonstrated in this manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The paper does what it says: it interpolates the Asai–Flach Euler system of [LLZ18] in Hida families for real quadratic fields, with explicit specialization factors, and this is already an input for [GLZ25] on Bloch–Kato in analytic rank zero. The novelty is not the broad outline—that follows the Beilinson–Flach/Rankin–Selberg playbook—but the execution: a control theorem for nontrivial central character (Theorem A), the class z∞ on the G tower obtained by pushing forward from G*, and the exact interpolation factors in Theorem C. The paper is cleanly written, the convention choices are spelled out, and the inert-prime case of the p-stabilization relation is proved directly from [LLZ18]. The compatibility of moment maps (Prop. 5.3.1) is proved rather than black-boxed.\n\nThe soft spots are real but localized. Theorem A and Theorem 3.2.1 are delegated to 'proved in exactly the same way as [She25]'; that is likely fine given the argument for Theorem 3.1.5, but it does make the paper not fully self-contained at the foundational level. The bigger concern is the split-prime p-stabilization relation (Theorem 4.1.3), which is proved in Section 6.2 via a 'motivic' functional Z_mot,j. Proposition 6.2.1 asserts the existence and normalization of this functional, with a proof that is essentially a citation to [LZ24] and [Gro20] plus 'carefully keeping track of all the choices'. Then Lemma 6.2.2 identifies the space of such functionals with the Rankin–Selberg zeta integral, and the spherical normalization Z_an(W_sph, ch(Z_p^2))=1 fixes the proportionality constant. If any of these normalizations is off by a unit, the scalar factors in Theorem C are off by that unit. I found no actual error here, but this is a load-bearing dependency, and it is not actually proved in the manuscript. The stress-test note is right to focus on this: it is the only place where the advertised compatibility could silently break.\n\nBottom line: this deserves a serious referee. The referee should ask for a complete proof of Proposition 6.2.1, or at least a precise statement with a full proof in a cited companion, and check that the identification with Z_an,j is not just up to an unabsorbed constant. Once that piece is verified, this is a strong contribution. I would bring it to a reading group only if the group works on p-adic Euler systems; it is quite specialized. I would cite it if I worked on Asai representations or Hilbert modular forms.","headline":"A solid, important interpolation result whose main caveat is a deferred proof of the split-prime p-stabilization relation, not the overall strategy.","tokens_in":15115,"tokens_out":5476,"would_cite":true,"duration_ms":48576,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F33","11F41","11F80","11R23","14G35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Asai–Flach Euler system varies p-adically in Hida families, with the classical classes recovered as explicit scalar specializations of one Iwasawa class.","keywords":["Asai representation","Euler systems","Hida families","Hilbert modular forms","p-adic interpolation","Iwasawa cohomology","p-stabilization","Bloch–Kato conjecture"],"falsifier":"For a concrete real quadratic field, a split prime $p$, and a concrete ordinary Hilbert modular form of known Hecke eigenvalues, compute the specialization of $c_{\\mathrm{AF}}(\\Pi)$ at an algebraic weight and compare the scalar with the formula in Theorem 5.4.3; a discrepancy by a non-unit in the p-stabilization factor $R(p^{-1-h})$ would falsify the interpolation claim.","tokens_in":14054,"feed_emoji":"🧮","tokens_out":16297,"duration_ms":147270,"temperature":0.7,"pith_summary":"This paper claims that the Asai–Flach Euler system for Hilbert modular forms over a real quadratic field is p-adically interpolable: as the eigenform moves in a Hida family (a p-adic analytic family of ordinary Hilbert modular forms), the whole compatible system of cohomology classes can be packaged into one Iwasawa cohomology class. The main result produces a family of Galois representations $M(\\Pi)^*$, free of rank $4$ over the weight space, whose specializations are the Asai representations of the individual forms, and a class $c_{\\mathrm{AF}}(\\Pi)$ in $H^1_{\\mathrm{Iw}}(\\mathbb{Q}(\\mu_{p^\\infty}),M(\\Pi)^*)$ that specializes to the classical Asai–Flach classes multiplied by explicitly known scalars. A reader would care because this turns a collection of classes attached to isolated eigenforms into a single p-adic analytic object that follows the deformation, and the paper states that this object is the input used in proofs of the Bloch–Kato conjecture in analytic rank zero for Asai representations. The paper also establishes a derived control theorem for ordinary Iwasawa cohomology of Hilbert modular varieties with nontrivial central character.","feed_headline":"Asai–Flach Euler system interpolates p-adically in Hida families","feed_subtitle":"One Iwasawa class over the family specializes to each Asai–Flach class up to a known scalar.","key_machinery":"The machinery has three layers. First, the ordinary Iwasawa cohomology module $e'_{\\mathrm{ord}}H^*_{\\mathrm{Iw}}(Y_G(K_\\infty)_{\\overline{\\mathbb{Q}}},\\mathcal{O})$ over $\\Lambda=\\mathcal{O}[[S]]$, where $S$ is the diagonal-torus quotient, together with moment maps $\\mathrm{mom}^\\lambda_n$ into étale cohomology with coefficient sheaves $\\mathcal{H}[\\lambda]$; Theorem A's Tor spectral sequence describes exactly when these maps are isomorphisms. Second, the class $z_\\infty$, obtained by pushing forward the Asai–Flach class of [LLZ18] from the $G^*$-Shimura variety to the $G$-Shimura variety; projecting to an ordinary Hecke eigenspace gives $c_{\\mathrm{AF}}(\\Pi)$. Third, the p-stabilization relation (Theorem 4.1.3), whose Euler factor $R(X)$ records the change of the Asai–Flach class under p-refinement; the proof of that relation in the split-prime case identifies the motivic functional $Z_{\\mathrm{mot},j}$ with the analytic Rankin–Selberg functional $Z_{\\mathrm{an},j}$ using the one-dimensionality of the space of equivariant linear forms.","core_discovery":"On the paper's own terms, the central discovery is Theorem C: for an ordinary family $\\Pi$ over an affinoid disc $\\mathcal{C}$ in weight space, there is a Galois representation $M(\\Pi)^*$ over $\\mathcal{C}$, free of rank $4$ and equipped with a Hecke action, whose specialization at every algebraic weight $\\lambda\\in\\mathcal{C}$ is canonically isomorphic to the Asai representation attached to the specialized Hilbert modular form $\\Pi[\\lambda]$. With it comes an Iwasawa cohomology class $c_{\\mathrm{AF}}(\\Pi)\\in H^1_{\\mathrm{Iw}}(\\mathbb{Q}(\\mu_{p^\\infty}),M(\\Pi)^*)$ whose evaluation at $(\\lambda,h)$ is an explicit scalar multiple of the prime-to-$p$ Asai–Flach class $\\mathrm{AF}[\\Pi,j]_{\\acute{e}t}$; the scalar is built from the p-stabilization Euler factor $R(p^{-1-h})$ of Theorem 4.1.3. The paper also proves the intermediate Theorem B, which realizes the Asai–Flach elements in ordinary Iwasawa cohomology, and Theorem A, a Tor spectral sequence controlling specializations of ordinary cohomology with general central character.","pith_inferences":["A direct numerical check on a small real quadratic field and a split prime would verify the p-stabilization scalar $R(p^{-1-h})$, effectively testing the one-dimensionality argument without building a new theory.","The same derived control theorem should apply to any Shimura-variety Euler system whose coefficient sheaves have central character through the norm map, so the interpolation mechanism is probably not specific to Asai representations.","The $(\\lambda,h)$ redundancy identified in Remark 5.4.4 suggests a cleaner formulation: fix the central-character weight and let only the cyclotomic variable move, which may make the family class directly comparable with p-adic L-functions.","If the ordinary projector is replaced by a nearly-ordinary one, the construction should interpolate p-stabilized classes for forms that are only nearly ordinary at $p$, as hinted in Remark 3.3.4."],"forward_implications":["Every classical Asai–Flach class in the family is a specialization of one global class, so arithmetic invariants computed from the Euler system can be studied as functions on weight space.","The explicit scalar $R(p^{-1-h})$ pins down how the class changes when the level is p-stabilized, which is what lets the interpolated class be compared with the prime-to-$p$ classes of [LLZ18].","The rank-4 family $M(\\Pi)^*$ gives a deformation of the four-dimensional Asai Galois representations over the weight disc, the object needed for Selmer-group and Bloch–Kato arguments over the family.","The control theorem recovers exact control of ordinary cohomology at maximal ideals with non-solvable residual image under weaker hypotheses than previous results, and applies to coefficient systems with nontrivial central character.","According to the paper, this interpolated Euler system is a required input for the proof of the Bloch–Kato conjecture in analytic rank zero for the Asai representation."],"supporting_citations":[{"why":"Defines the Asai–Flach Euler system and the classes whose p-adic interpolation is the paper's subject.","marker":"[LLZ18]"},{"why":"Provides the control theorems and level conventions for Hilbert modular varieties that Theorem A extends.","marker":"[She25]"},{"why":"Supplies the norm-relation comparison behind the motivic Asai–Flach functional used in the split-prime p-stabilization proof.","marker":"[Gro20]"},{"why":"Provides the template for tracking choices in the construction of the motivic functional $Z_{\\mathrm{mot},j}$.","marker":"[LZ24]"},{"why":"Proves the one-dimensionality of equivariant linear forms used to identify the motivic and analytic functionals in Lemma 6.2.2.","marker":"[HS01]"},{"why":"Supplies the Whittaker-model and multiplicity-one machinery used for the split-prime case of Theorem 4.1.3.","marker":"[LSZ22]"},{"why":"Gives the cohomological vanishing used to make the ordinary control theorem exact at suitable maximal ideals.","marker":"[CT23]"}],"fun_headline_variants":["One p-adic class specializes to every Asai–Flach class","Asai–Flach Euler system has p-adic family interpolation","p-adic families of Asai–Flach Euler systems","Interpolating Asai–Flach classes over weight space"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the geometric construction of the Asai–Flach class and the analytic Rankin–Selberg integral computation agree exactly, with no missing constant; if that constant were wrong, the explicit scalar factors in the main theorem would no longer match the classical classes.","fun_headline_variants_meta":{"raw":{"variants":["One p-adic class specializes to every Asai–Flach class","Asai–Flach Euler system has p-adic family interpolation","p-adic families of Asai–Flach Euler systems","Interpolating Asai–Flach classes over weight space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3412,"prompt_tokens":885,"completion_tokens":2527,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":2465}},"tokens_in":501,"tokens_out":2527,"duration_ms":18044,"temperature":1.0,"reasoning_tokens":2465,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:28:31.714536+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete real quadratic field, a split prime $p$, and a concrete ordinary Hilbert modular form of known Hecke eigenvalues, compute the specialization of $c_{\\mathrm{AF}}(\\Pi)$ at an algebraic weight and compare the scalar with the formula in Theorem 5.4.3; a discrepancy by a non-unit in the p-stabilization factor $R(p^{-1-h})$ would falsify the interpolation claim.","supporting_citations":[],"review_version":2}