{"id":"53d5777e-1163-4c28-9356-25d72b3d457c","arxiv_id":"2412.18881","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines ordinary distributions on Rankin-Selberg eigenvarieties and proves that the Bessel period divisor controls the Selmer group divisor in one direction.","lead":"This paper constructs p-adic analytic families of Bessel periods and Selmer groups over an eigenvariety for unitary groups, and proves one divisibility direction of an Iwasawa-type main conjecture linking them. It matters because it lifts Iwasawa theory from single anticyclotomic twist directions to the full space of ordinary automorphic deformations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 11.5 is conditional on the unproved [LTXZZ1, Hypothesis 2.2.5] for all dominant weights; if this hypothesis fails, the ordinary eigenvariety and the Galois module R_J(ξ,V)_gamma may not exist or lack the required properties.","rationale":"The reader's verdict of CONDITIONAL is exactly right. The paper is a substantial and internally coherent contribution, and the author honestly discloses the main external dependencies: the unproved [LTXZZ1, Hypothesis 2.2.5] for all dominant weights and the unpublished [LSb]. My stress-test pass did not find an internal contradiction or a clear mathematical error in the proof structure; the central claim is conditional precisely because these external inputs are load-bearing. In particular, the construction of the ordinary eigenvariety, the identification of the Galois representation R_J(xi,V)_gamma, and the reciprocity laws in Sections 9 and 10 all rely on these inputs. Since the reader already marked the verdict CONDITIONAL with the same weakest assumption, no change is needed. A stronger verdict (ACCEPT) would be inappropriate without public availability of [LSb] and a proof of the hypothesis. A REJECT would also be inappropriate because the paper explicitly states the conditional nature and does not claim an unconditional theorem.","tokens_in":75526,"tokens_out":9337,"duration_ms":90522,"concrete_test":"Check the current status of [LTXZZ1, Hypothesis 2.2.5]: determine whether its proof for the minimal weight has been extended to all dominant weights, or whether a counterexample is known for F^+ = Q and n > 2. Simultaneously, verify whether the manuscript [LSb] by D. Liu and B. Sun has been made public or replaced by a published reference; if either remains unavailable, the theorem should be restated as an explicit conditional result and the verdict should remain CONDITIONAL.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central result (Theorem 11.5) is explicitly stated in Remark 1.4 to be conditional on [LTXZZ1, Hypothesis 2.2.5] holding for all dominant weights for N = n, n+1. This hypothesis concerns the existence and control of Galois representations and R = T theorems in the automorphic setting; it is used, for example, in Lemma 4.14 (via [Sch18] and Remark 1.4) to identify D^flat_J(xi,V)_gamma with a module over R_J(xi,V)_gamma, and in the construction of the ordinary eigenvariety itself. The author notes the results are still conditional when F^+ = Q and n > 2. Since the hypothesis is not proved in this paper or in a publicly available reference cited here, the entire bipartite Euler system argument rests on an unverified external input. Additionally, the foundational construction of ordinary distributions (Section 3 and Section 4) and the explicit reciprocity laws (Sections 9 and 10) rely on the unpublished manuscript [LSb] (in preparation); if [LSb] is unavailable or contains gaps, the definitions of D_J(xi,V)_gamma and lambda_J(V) are not fully justified. This is a genuine load-bearing concern, distinct from an internal inconsistency in the argument; it prevents full verification of the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a notion of ordinary distributions for unitary groups and their Rankin--Selberg products, uses it to (re)construct ordinary eigenvarieties in both definite and indefinite settings, and defines Bessel periods as elements of these distribution spaces (definite case) or of Selmer groups (indefinite case). It then formulates an Iwasawa-type main conjecture over the ordinary Rankin--Selberg eigenvariety relating the vanishing divisor of the Bessel period to the characteristic divisor of a Selmer group, and proves one side of the divisibility under Assumption 4.7 plus Fontaine--Laffaille and interlacing conditions (Theorem 11.5; variant Theorem 11.6). The proof builds a bipartite Euler system over the eigenvariety using two explicit reciprocity laws and an R=T theorem for ordinary distributions.","tokens_in":75915,"tokens_out":9425,"duration_ms":89608,"significance":"If the result is correct, this is a substantial extension of the Iwasawa theory of [LTX] and [LTXZZ1] from anticyclotomic twists to the full ordinary eigenvariety, and the framework of ordinary distributions is a new and potentially useful tool. The paper is commendably explicit about its conditionalities: the main theorem is conditional on [LTXZZ1, Hypothesis 2.2.5] and on the unpublished manuscript [LSb], and many key lemmas are delegated to 'the same argument as' in prior work. The construction of the Bessel period is independent of the Selmer side, and no parameter is fitted, so there is no circularity in the argument. However, because the foundational distribution spaces and the reciprocity laws depend on unavailable or unproved external inputs, the paper does not yet provide a fully verifiable proof of its main theorem.","major_comments":[{"comment":"The central objects D_J(ξ_N,V_N,K_N) and the ordinary eigenvariety are built from the poset k_I and the transfer maps recalled from the unpublished manuscript [LSb, §2.5]. Since [LSb] is not available, the existence and properties of these distributions, including the cofinality Lemma 3.4 and the weight-independence arguments in Proposition 3.8, cannot be independently verified. This is load-bearing: the later Bessel period λ_J(V) of §6 and the main theorem concern exactly these D_J. The author should either incorporate the necessary statements and proofs from [LSb] into the paper or make them precise and explicit assumptions.","section":"§3, Construction 3.2 and Definition 3.6"},{"comment":"The Galois module R_J(ξ,V)_γ of Remark 4.12 and the eigenvariety E_J(ξ,V)_γ are constructed under [LTXZZ1, Hypothesis 2.2.5] for all dominant weights N=n,n+1. Lemma 4.14 explicitly invokes this hypothesis via [Sch18]. If Hypothesis 2.2.5 fails, the module R_J may not exist and the Selmer groups of §5 are undefined, so Theorem 11.5 would have no content. The paper acknowledges this only in a remark; the theorem statements should carry the hypothesis explicitly rather than relegating it to Remark 1.4.","section":"Remark 1.4 and Lemma 4.14"},{"comment":"The first explicit reciprocity law, which is a key input to the bipartite Euler system of §13, is proved by asserting that the arguments of [LTXZZ1, Lemma 6.2.2, Theorem 6.2.3, Proposition 6.3.1, Theorem 6.3.4, Lemma 7.2.5, Theorem 7.2.8, Proposition 7.2.7] carry over 'in the same way' to ordinary distributions over the eigenvariety with coefficients Z^∨_ξ/p^m. The necessary new ingredients—such as the behavior of the weight spectral sequence and Gysin maps for the nonconstant local system under the ordinary projector—are not presented. Since Theorem 13.13(1) depends on Theorem 9.7, this gap is load-bearing for the proof of Theorem 11.5.","section":"§9, Lemmas 9.2–9.5 and Theorem 9.7"},{"comment":"The second explicit reciprocity law is proved by indicating that the argument of [LTX, Theorem 4.3.6] (essentially [LTXZZ1, Theorem 4.6.2]) applies. In particular, the proof relies on the invertibility of the Hecke operators (T^{⋆′}_{nα,l})_{rα} and on compatibility of the ordinary projection with the inverse limits defining D_J(ξ,V)_γ. These compatibilities are not shown in the manuscript. Since Theorem 10.1 is used directly in Theorem 13.13(2), this is a second load-bearing point in the Euler-system construction.","section":"§10, Theorem 10.1"},{"comment":"The final step from the pointwise statements of Proposition 14.2 to the divisibility of characteristic divisors uses Lemma 14.3, whose proof contains a claim about limits of lengths of Tor modules that is only sketched ('for the claim, we may assume that φ is either injective or surjective...'). In addition, the appendix R=T theorem (Theorem 15.2) is asserted by 'running the same argument' as [LTXZZ2, Theorem 3.38] after replacing the base ring; the global Taylor–Wiles patching with the filtered local deformation problems D^fil_v is not carried out. Since Lemma 13.8 and Proposition 13.7 depend on Theorem 15.2, the bipartite Euler system is not fully verified without these details.","section":"§14, §15"}],"minor_comments":[{"comment":"The sentence 'In fact, We conjecture that...' has an erroneous capital 'W' after the comma; it should read 'we conjecture'.","section":"Remark 4.11"},{"comment":"The phrase 'we will only discuss the (much harder) case where n0 ≥ 4 (hence n1 ≥ 3) and leave the (much easier) case where n0 = 2 to the readers as an exercise' is too informal for a journal article; the n0=2 case either needs a proof or a precise reference.","section":"§9, after Definition 9.1"},{"comment":"The phrase 'for l ∈ L0 whose underlying rational prime is coprime to n' is ambiguous since n is a set of primes; it should say 'coprime to the product of the underlying rational primes of the elements of n'.","section":"Notation 13.1(2)"},{"comment":"In the displayed line 'H(Sh(G_N, K_N I), Z_{ξ_N}/p^m)' the notation 'K_N I' is not defined before; it would be clearer to write the open compact subgroup as a product or define it explicitly.","section":"Lemma 3.4(2)"},{"comment":"The phrase 'then one recovers [LTXZZ1, Theorem 1.1.1 & Theorem 1.1.7] (and their rank one analogues)' would benefit from stating whether the recovery is as an equality of divisors or only as the one-sided divisibility proved here.","section":"§1, Remark 1.3(1)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central results are conditional on the unpublished manuscript [LSb] and on the unproved [LTXZZ1, Hypothesis 2.2.5]. Before acceptance, the editor should decide whether a journal in number theory accepts a paper whose main theorem depends on an in-preparation manuscript; ideally [LSb] should be posted to arXiv or the relevant constructions should be incorporated into the present paper. The delegation of many proofs to [LTXZZ1] and [LTX] is in line with the author's previous work, but the number of such delegations is unusually high for the core reciprocity laws. The conceptual framework and the precise statement of the conjecture are valuable, and there is no evidence of circularity or parameter-fitting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is the real thing: it introduces ordinary distributions for unitary groups and their Rankin–Selberg products, uses them to redefine ordinary eigenvarieties, and proves a one-sided divisibility in an Iwasawa-type conjecture over the eigenvariety. That is a genuine step beyond the prior anticyclotomic results in [LTXZZ1] and [LTX], not a repackaging. The Bessel period construction and the bipartite Euler system over the eigenvariety are new, and the paper recovers the earlier theorems as special cases. Credit where due: the author is explicit about what is conditional, the internal structure is coherent, and the appendix proves a real R = T theorem for ordinary distributions.\n\nNow the soft spots, in proportion. The main theorem rests on two external inputs: [LTXZZ1, Hypothesis 2.2.5] (existence/control of Galois representations, R = T in the relevant setting) and the unpublished manuscript [LSb] for the foundational Construction 3.2 and the explicit reciprocity laws. The author states this plainly in Remark 1.4, and notes the results are still conditional when F+ = Q and n > 2. That is load-bearing, but it is not hidden or misrepresented. The paper also delegates several lemmas to “the same argument as” in prior work; that is normal in this area but makes verification genuinely harder. The theorem proves only the divisibility, not the equality in Conjecture 11.2, and the tempered locus may be strictly smaller than the normal locus—again, stated in Remark 11.4.\n\nI do not think these issues warrant rejection. The framework is coherent, the limitations are acknowledged, and the conditional nature is structural to the current state of the program. The main worry for a referee is whether [LSb] is actually available or likely to appear; if it is not, the foundational definitions are not fully justified. That is a real concern, but it is a tractable one. The paper deserves a serious referee and probably a conditional accept, with the condition being that the dependencies are resolved or clearly documented.\n\nFor your reading group: yes, if anyone works on Iwasawa theory or eigenvarieties. I would cite it if I worked in the area, and I would send it out.","headline":"A substantial, honest conditional upgrade of the anticyclotomic Iwasawa main conjecture to the eigenvariety level; the main theorem is clearly flagged as resting on an unpublished manuscript and an unproved hypothesis.","tokens_in":76356,"tokens_out":1536,"would_cite":true,"duration_ms":18934,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F33","11G05","11G18","11G40","11R34"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves an eigenvariety-level Iwasawa divisibility: where the Bessel period is nonzero on a component, the Selmer group vanishes or is torsion, and its characteristic divisor is bounded by twice the period's vanishing divisor.","keywords":["ordinary distributions","ordinary eigenvariety","Rankin-Selberg","Bessel period","Selmer groups","Iwasawa main conjecture","characteristic divisor","unitary groups"],"falsifier":"A single interlacing classical point on a tempered component where the Bessel period is nonzero but the Selmer group has positive rank would disprove Theorem 11.5(1a); equivalently, computing the two divisors at a codimension-one point and finding $2\\,\\mathrm{char}_{E'}(D_J^H/\\lambda_J(V))-\\mathrm{char}_{E'}(X_J)$ not effective would disprove the divisibility.","tokens_in":75295,"feed_emoji":"🧮","tokens_out":13976,"duration_ms":115310,"temperature":0.7,"pith_summary":"This paper lifts Iwasawa's main conjecture from a single anticyclotomic tower to the whole ordinary eigenvariety attached to a Rankin–Selberg product of unitary groups of ranks $n$ and $n+1$. The author introduces 'ordinary distributions'—the continuous duals of nearly ordinary automorphic forms—and uses them to rebuild the ordinary eigenvariety. The Bessel period is constructed as a canonical element of this distribution space in the definite case, and as an element of a Selmer group (the Galois-cohomology classes satisfying specified local conditions) in the indefinite case. The conjecture pairs the vanishing divisor of the period with the characteristic divisor of the Selmer group of the associated Galois module, with a factor of $2$; the paper proves one side of the divisibility. Under the stated hypotheses, the paper proves the control side of the conjecture: in the definite case nonvanishing of the period makes the Selmer group vanish and the dual Selmer module torsion, while in the indefinite case it forces generic rank one and the same divisibility for the torsion part.","feed_headline":"Bessel period zeros bound Selmer groups over an eigenvariety","feed_subtitle":"Proves one side of an Iwasawa-type conjecture: where the period does not vanish, the Selmer group collapses.","key_machinery":"The engine of the paper is the space $D_J(\\xi,V)_\\gamma$ of $J$-invariant ordinary distributions: the inverse limit, over open compact subgroups, of continuous duals of nearly ordinary automorphic forms on $U(n)\\times U(n+1)$, whose endomorphism ring $E_J(\\xi,V)_\\gamma$ cuts out the integral ordinary eigenvariety $\\mathrm{Spec}\\,E_J(\\xi,V)_\\gamma$. The Bessel period $\\lambda_J(V)$ is a compatible family of period sums over the diagonal subgroup $H$, interpolating the classical Bessel periods at interlacing classical points; in the indefinite case the analogue is a Selmer-class $\\kappa_J(V)$. The Selmer module $X_J(\\xi,V)_\\gamma$ is the Pontryagin dual of the Selmer group of the Rankin–Selberg Galois module $R_J(\\xi,V)_\\gamma$, a module of rank $n(n+1)$ interpolating the standard Galois representations of the two unitary groups. The comparison invariant is the characteristic divisor, a sum of lengths at codimension-one points. The proof is carried by a bipartite Euler system—a graph of congruence modules over sets of auxiliary primes joined by two explicit reciprocity laws—which matches the period with Selmer classes at level-raising primes.","core_discovery":"The central result, Theorem 11.5, is a divisibility between two codimension-one invariants on the ordinary eigenvariety. For a residual Galois representation satisfying the paper's Assumption 4.7, an interlacing Fontaine–Laffaille regular weight (ordering and $p$-smallness conditions on the weights that make the Bessel period well defined and the local representations crystalline), and $p > 2(n_0+1)$, the paper proves that on an irreducible component $E'$ of the tempered locus (an open dense subset of the normal locus containing all classical points), nonvanishing of the Bessel period $\\lambda_J(V)$ implies (a) the Selmer group $H^1_f(F,R_J(\\xi,V)_\\gamma)$ vanishes over $E'$, (b) the dual Selmer module $X_J(\\xi,V)_\\gamma$ is torsion over $E'$, and (c) the divisor $2\\,\\mathrm{char}_{E'}(D_J(\\xi,V)_\\gamma^H/\\lambda_J(V))-\\mathrm{char}_{E'}(X_J(\\xi,V)_\\gamma)$ is effective. An analogous statement holds in the indefinite case: nonvanishing of the cohomological period class $\\kappa_J(V)$ forces generic rank one for the Selmer group and its dual, and gives the corresponding effective divisibility for the torsion part of the dual Selmer module. The proof proceeds through two explicit reciprocity laws that relate the period to Selmer classes at auxiliary primes, organized into a bipartite Euler system over the eigenvariety, and then specializes to closed points to compare lengths and divisors.","pith_inferences":["If the opposite divisibility were established, the vanishing divisor of the Bessel period would exactly compute the characteristic divisor of the Selmer group along each component, turning the period into a p-adic regulator on the eigenvariety.","The same ordinary-distribution formalism could be applied to other period functionals of the same shape, yielding analogous eigenvariety-level Selmer divisibilities.","A testable strengthening is to prove that the tempered locus equals the full normal locus in the cases the paper notes where their difference is empty, which would remove the one geometric restriction in the theorem.","Removing the unproved standing hypothesis would make the results unconditional over the rational base field when $n>2$, since the author states the conditionality arises precisely from that input."],"forward_implications":["In the definite case, nonvanishing of the Bessel period over a tempered component forces the Selmer group of the Rankin–Selberg Galois module to vanish over that component.","In the indefinite case, the same nonvanishing forces the Selmer group and its dual to have generic rank one over the component, with the torsion part of the dual controlled by the period divisor.","The theorem recovers the earlier anticyclotomic Iwasawa main conjecture and the earlier rank-zero and rank-one results as special cases, now for arbitrary interlacing Fontaine–Laffaille regular weights and without requiring the deformation places to split.","The proven divisibility is the upper-bound half of the proposed main conjecture: it shows the Selmer group cannot be larger than what the vanishing of the Bessel period allows.","The paper's ordinary-distribution definition of the eigenvariety works without choosing an open compact subgroup at the p-adic deformation places, which may simplify future constructions."],"supporting_citations":[{"why":"Supplies the standing Hypothesis 2.2.5, the level-raising and first explicit reciprocity law that the paper upgrades to the eigenvariety, and the earlier rank-zero and rank-one theorems recovered as special cases.","marker":"[LTXZZ1]"},{"why":"Establishes the anticyclotomic Iwasawa main conjecture and the specialization and length-comparison lemmas that the present proof adapts to a full eigenvariety.","marker":"[LTX]"},{"why":"Defines the Selmer groups and local conditions that the paper uses for the Galois module over the eigenvariety.","marker":"[BK90]"},{"why":"Provides the relative completed cohomology and modular-symbol framework used to construct the Bessel period as an ordinary distribution.","marker":"[LSb]"},{"why":"Supplies the local restriction result used to show the period interpolates classical Bessel periods and is nonzero at the relevant classical points.","marker":"[BPLZZ21]"},{"why":"Provides the universal deformation ring and R=T results that the appendix extends to ordinary distributions, and the key result used in level raising.","marker":"[LTXZZ2]"},{"why":"Supplies the Galois deformation rings and local deformation problems used throughout the appendix R=T theorem.","marker":"[CHT08]"}],"fun_headline_variants":["Period vanishing forces Selmer collapse over eigenvariety","Eigenvariety divisibility: period zeros vs Selmer torsion","Nonvanishing Bessel period annihilates Selmer group","Selmer vanishes where Bessel period persists: Iwasawa clue","One-sided Iwasawa: period zeros divide Selmer characteristic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is an unproved standing hypothesis, stated as Hypothesis 2.2.5 in the companion paper [LTXZZ1], that for both unitary groups in the pair the relevant automorphic Galois representations exist and are controlled by the Hecke algebra at every dominant weight; the author notes the results are conditional when the base field is $\\mathbb{Q}$ and $n>2$.","fun_headline_variants_meta":{"raw":{"variants":["Period vanishing forces Selmer collapse over eigenvariety","Eigenvariety divisibility: period zeros vs Selmer torsion","Nonvanishing Bessel period annihilates Selmer group","Selmer vanishes where Bessel period persists: Iwasawa clue","One-sided Iwasawa: period zeros divide Selmer characteristic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1404,"prompt_tokens":991,"completion_tokens":413,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":327}},"tokens_in":607,"tokens_out":413,"duration_ms":3610,"temperature":1.0,"reasoning_tokens":327,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:21:51.998151+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A single interlacing classical point on a tempered component where the Bessel period is nonzero but the Selmer group has positive rank would disprove Theorem 11.5(1a); equivalently, computing the two divisors at a codimension-one point and finding $2\\,\\mathrm{char}_{E'}(D_J^H/\\lambda_J(V))-\\mathrm{char}_{E'}(X_J)$ not effective would disprove the divisibility.","supporting_citations":[],"review_version":1}