{"id":"5a96de59-f991-48d7-bfe2-3ca1d53a5585","arxiv_id":"2412.11147","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Rankin-Selberg products of Hida and Coleman families, the paper proves equality of characteristic ideals of Selmer groups under the involution, yielding functional equations for algebraic p-adic L-functions.","lead":"This paper proves functional equations for algebraic p-adic L-functions attached to products of modular form families, relating the Selmer side of the L-function at a character to the dual character. The argument works by identifying these L-functions with determinants of Selmer complexes, which is a reusable tool for non-ordinary and multi-variable settings in Iwasawa theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 2-parameter non-ordinary functional equation is conditional on an unproved Tamagawa projectivity hypothesis (Tam_V); until H^1(I_q,V) and H^1(I_q,V*) are shown free over A for q∤p, Theorem 6.20 does not establish the promised parity removal.","rationale":"The reader's weakest assumption is exactly the point I find load-bearing. In Part 3, Theorem 6.19 assumes (Tam_V) rather than proving it, and Theorem 6.20 then invokes Theorem 5.18/5.25, whose projectivity condition is the same unverified hypothesis. In the ordinary Part 1, the paper supplies a reduction (Propositions 2.6, 2.8, 2.9) to a one-point Tamagawa check, making the condition concrete; no such reduction is offered for non-ordinary 2-parameter Coleman families. The tautological first equality in the proof of Theorem 6.20 and the self-citation in the proof of Theorem 6.23 reinforce that the family-level hypotheses are not fully checked. This does not mean the main theorems are false; it means the advertised unconditional removal of the parity condition is not yet supported. I therefore do not propose changing the reader's CONDITIONAL verdict; the concern confirms that the paper is a conditional contribution whose key arithmetic hypothesis still needs verification.","tokens_in":46951,"tokens_out":8196,"duration_ms":79335,"concrete_test":"Analytical check: attempt to prove the Coleman-family analogue of Proposition 2.8 for the 2-parameter case. Specialize V = V_f^* ⊗ V_g^* to a single pair of arithmetic points, assume the p-part of the Tamagawa factor at q is 1 for that specialization, and try to show H^1(I_q,V) is free over A = E<X1,X2> by imitating the proof of Proposition 2.8 with [BCS23, Props 4.25/4.26]. If the proof succeeds, (Tam_V) becomes a checkable one-point condition and Theorem 6.20 is unconditional up to that check; if the proof fails at the freeness step, Theorem 6.19/6.20 are missing an essential verification and the parity-removal claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing point is the Tamagawa condition (Tam_V) in Theorem 6.19: H^1(I_q,V) and H^1(I_q,V*) are assumed projective A-modules for q in Sigma_p. Theorem 6.20 passes to the functional equation through Theorem 5.18(iii)/5.25, and that theorem requires exactly this projectivity for all q in S(p). For A = E<X1,X2>, a two-dimensional regular affinoid algebra, freeness is not automatic: H^1(I_q,V) can contain torsion, and if it does the Selmer complex is not perfect with amplitude [1,2], so det(RGamma) = char fails and the functional equation does not follow. In the ordinary case the paper supplies a concrete reduction: Propositions 2.8 and 2.9 show that vanishing of the p-part of the Tamagawa factor at one arithmetic specialization implies freeness of H^1(I_v,T2) over R2 and vanishing of the error terms, via [BCS23, Props 4.25/4.26]. No analogous reduction is proved for the non-ordinary 2-parameter Coleman families of Section 6.3.2; Theorem 6.19 simply says 'Suppose (Tam_V) holds.' The proof of Theorem 6.20 also contains a tautological first equality (the same determinant appears on both sides), and the proof of Theorem 6.23 cites 'Theorem 6.23' itself, which should presumably be Theorem 6.22. These are symptoms that the family-level hypotheses are not fully verified. The main advertised consequence, removal of the parity condition in earlier Iwasawa main-conjecture results, therefore remains conditional on a hypothesis that the paper does not prove or reduce to a checkable condition in the non-ordinary setting.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an abstract framework, following Nekovář and Benois, in which algebraic p-adic L-functions attached to families of Galois representations are identified with determinants of Selmer complexes, and functional equations are derived from duality. The main abstract result is Theorem 5.25, which asserts an equality of characteristic ideals of degree-2 Selmer cohomology under hypotheses of perfectness, degree-2 concentration, torsion, and projectivity of local H^1(I_q,V). This is applied first to ordinary Hida families (Theorems 2.15 and 2.21) and then to Rankin–Selberg products of non-ordinary Coleman families (Theorems 6.10, 6.16, 6.20, and 6.22–6.23), with the advertised application of removing parity conditions in earlier Iwasawa main-conjecture results.","tokens_in":47242,"tokens_out":5351,"duration_ms":50082,"significance":"If the main theorems are correct, the paper would give a uniform determinant-theoretic proof of algebraic functional equations for Rankin–Selberg p-adic L-functions, and would strengthen existing Iwasawa main-conjecture results by eliminating parity restrictions. The paper has clear strengths: it works in substantial generality over affinoid algebras and their distribution rings, it gives a careful comparison of determinants and characteristic ideals for coadmissible modules (Proposition 3.5), and it reduces several perfectness questions to Tamagawa-type conditions, with an explicit and useful analysis in the ordinary case (Propositions 2.6, 2.8, 2.9). The reliance on external duality results (Nekovář, Benois, Pottharst, Küçük) is transparent, and the control-theorem arguments in Section 6.3 are generally carefully presented. However, as discussed in the major comments, the main non-ordinary 2-parameter result is conditional on an unproved projectivity hypothesis, and several load-bearing proof steps contain gaps or tautological statements that must be repaired before the advertised parity-removal claim is established.","major_comments":[{"comment":"The first displayed equality in the proof of Theorem 6.20 is tautological: the same expression det_{HA(Γ_Q^0)} RΓ(V*(1), D⊥) appears on both sides. The theorem statement itself also writes char of RΓ(V,D) and RΓ(V*(1),D⊥), although characteristic ideals are defined only for modules, not for complexes. The intended argument presumably is to use Theorem 5.18 to obtain det RΓ(V*(1),D⊥) = det RΓ(V,D)^ι and then to identify these determinants with char R^2Γ(...) via Theorem 5.24. As written, the proof does not supply the key duality step that yields the equality of determinants.","section":"§6.3.2, Theorem 6.20"},{"comment":"The Tamagawa condition (Tam_V), namely projectivity of H^1(I_q,V) and H^1(I_q,V*) over A for q ∈ Σ_p, is simply assumed in Theorem 6.19 and is never verified or reduced to a checkable condition. This hypothesis is load-bearing: Theorem 5.18(iii) and Theorem 5.20 require exactly this projectivity for all q ∈ S(p) to obtain the duality isomorphism and perfectness used in Theorem 6.20. For A = E⟨X_1,X_2⟩, a two-dimensional regular affinoid algebra, projectivity is not automatic, and the analogues of Propositions 2.8 and 2.9, which in the ordinary case reduce the condition to vanishing of the p-part of a Tamagawa factor at one arithmetic specialization, are not proved in the non-ordinary 2-parameter setting. Thus the parity-removal consequence advertised in the introduction remains conditional on a hypothesis that the paper does not establish.","section":"§6.3.2, Theorem 6.19"},{"comment":"Proposition 5.22 provides a resolution by finitely generated projective HA(Γ_F^0)-modules of the same rank, but Theorem 5.24 and Theorem 5.25 replace these by free modules HA(Γ_F^0)^⊕r. Proposition 3.5, which is used to identify det with char, assumes a resolution by free modules. No argument is given that finite projective modules over HA(Γ_F^0) are free, and this is not automatic in general. This gap affects the central determinant-to-characteristic-ideal identification on which the functional equations depend.","section":"§5.7–5.8, Proposition 5.22, Theorems 5.24 and 5.25"},{"comment":"Proposition 5.17 states that the map (5.14) is a quasi-isomorphism if and only if (5.14) is. This is vacuous as written and cannot serve as a lemma in the proof of Theorem 5.18. Either the statement has a typo (for example, one side should involve a dual or a different local condition) or it should be removed and replaced by the concrete local duality statement needed in Theorem 5.18(ii).","section":"§5.6.6, Proposition 5.17"},{"comment":"The proof of Theorem 6.23 says that the hypotheses of Theorem 5.25 hold 'thanks to Theorem 6.23', which is a self-citation. Since Theorem 6.23 is the statement being proved, the reference should presumably be to Theorem 6.22, which is cited from Küçük. As written, this is a circular proof, although the intended correction is local.","section":"§6.4, Theorem 6.23"}],"minor_comments":[{"comment":"The sentence 'In this subsection, where we closely follow closely follow [LZ16a]' contains a duplicated phrase 'closely follow closely follow'; it should be corrected.","section":"§2.5"},{"comment":"'Hece field' should be 'Hecke field'.","section":"Assumption 2.7(3)"},{"comment":"'of of Krull-dimension 3' contains a duplicated 'of'.","section":"§2.1.2"},{"comment":"In part (i) of the proof, the vanishing R^3Γ(V_k,D_k) = 0 is attributed to Theorem 6.9(ii), but the relevant statement is Theorem 6.9(i).","section":"§6.3.2, Theorem 6.19 proof"},{"comment":"The height-0 prime case is dismissed with 'can be proved by mimicking the argument employed in the previous part'; since the prior argument uses the regular sequence {℘′,℘} and a maximal ideal, the height-0 case deserves at least a sentence indicating which argument is mimicked.","section":"§6.3.1, Proposition 6.14"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution with a plausible overarching strategy, and the ordinary-case analysis and the abstract determinant-versus-characteristic-ideal results are valuable. However, the 2-parameter non-ordinary functional equation is conditional on an unproved Tamagawa projectivity hypothesis, and the proofs of Theorems 6.20 and 6.23, as well as Proposition 5.17 and the free-resolution gap in Theorems 5.24–5.25, need substantive correction. These issues are within the scope of a major revision rather than a rejection, provided the authors either prove or explicitly relegate the Tamagawa condition to a stated assumption and repair the proof gaps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a substantial and mostly careful paper. The main line is right: recast algebraic p-adic L-functions as determinants of Selmer complexes, prove the determinant equals the characteristic ideal, and get the functional equation from duality. The determinant-characteristic ideal dictionary (Propositions 3.2 and 3.5) and the cylindrical characteristic ideal over HA(Γ_F^0) are genuinely new and useful. The ordinary case (Section 2) is complete, and the 1-parameter non-ordinary case (Theorem 6.16) appears to hold using known work of Küçük and the authors' own Proposition 6.14.\n\nThe soft spot is the 2-parameter non-ordinary case. Theorem 6.19 simply assumes (Tam_V), i.e. that H^1(I_q,V) and H^1(I_q,V*) are projective A-modules for q in Σ_p. For A = E⟨X1,X2⟩ that freeness is not automatic, and without it the Selmer complex need not be perfect with amplitude [1,2], so the determinant-equals-characteristic-ideal step fails. In the ordinary case the paper reduces this to a checkable Tamagawa-factor condition (Propositions 2.8–2.9), but no such reduction is given for non-ordinary 2-parameter Coleman families. The paper does flag the hypothesis, which is honest, but the abstract's phrase 'mild hypotheses' understates the matter: the headline consequence, removal of parity restrictions, is conditional on a hypothesis that may be hard to verify.\n\nThere are also two obvious typos that should be fixed: the proof of Theorem 6.20 begins with a tautological equality (the same determinant on both sides), and the proof of Theorem 6.23 cites 'Theorem 6.23' itself, presumably meaning Theorem 6.22. These are not mathematical errors, but they suggest the final section needs a careful editing pass.\n\nWho is this for? Iwasawa theorists working on p-adic L-functions and Euler systems. I would send it to a serious referee. The referee should ask the authors to either prove (Tam_V) in the non-ordinary setting or state clearly that the functional equation and the parity removal are conditional on it. With that addressed, the paper is a solid contribution worth publishing.","headline":"Solid determinant-based functional equations, but the advertised parity removal in the non-ordinary 2-parameter case rests on an unproved Tamagawa projectivity hypothesis.","tokens_in":47877,"tokens_out":3182,"would_cite":false,"duration_ms":28687,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R23"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a functional equation for algebraic p-adic L-functions of Rankin-Selberg products, via perfectness of the underlying Selmer complex and equality of its determinant with the characteristic ideal of the degree-2…","keywords":["Iwasawa theory","Rankin-Selberg products","p-adic L-functions","functional equations","Selmer complexes","Tamagawa factors","Coleman families","Hida families"],"falsifier":"Compute the module $H^{1}$(I_q,V) over the coefficient algebra A = O(X_1) ⊗ O(X_2) for an explicit pair of Coleman families at a prime q where a nontrivial Tamagawa factor is expected; the theorems require this module and its dual to be free, so exhibiting a single torsion element would refute the perfectness statement on which the functional equation rests.","tokens_in":46673,"feed_emoji":"🧮","tokens_out":8758,"duration_ms":79406,"temperature":0.7,"pith_summary":"The paper proves that the algebraic p-adic L-function attached to a Rankin-Selberg product of modular form families satisfies a functional equation: the characteristic ideal of the degree-2 Selmer group for the family equals, after the natural involution on the coefficient ring, the characteristic ideal for the dual family. The proof works by showing that the relevant Selmer complex is perfect with amplitude [1,2], that its cohomology is concentrated in degree 2, and that its determinant is exactly the characteristic ideal of the Selmer group. This makes the functional equation a formal consequence of duality for Selmer complexes rather than of explicit Cassels-Tate pairings. A key input is a condition on Tamagawa factors away from p, which guarantees that local duality maps are isomorphisms and that the determinant-to-characteristic-ideal step is legitimate. If the paper is right, earlier parity restrictions in Iwasawa main-conjecture results for symmetric squares and Rankin-Selberg products can be removed.","feed_headline":"Rankin-Selberg p-adic L-functions get a functional equation","feed_subtitle":"A determinant argument lets Iwasawa main conjectures for symmetric squares drop their parity assumptions.","key_machinery":"The machine is the Selmer complex RΓ(V,D) with coefficients in the rings HA($Γ_F^{0}$), built from a Tate algebra A and the cyclotomic Galois group $Γ_F^{0}$ ≅ Z_p. The load-bearing mechanism is the equivalence between two ways of forming an algebraic p-adic L-function: the determinant of this perfect complex equals the cylindrical characteristic ideal of its degree-2 cohomology, provided the complex is perfect with amplitude [1,2] and cohomology is concentrated in degree 2. The functional equation then follows from the cup-product duality pairing between RΓ(V,D) and RΓ(V^*(1),D^⊥), which becomes an isomorphism once local error terms vanish, and the involution ι on HA($Γ_F^{0}$) makes the determinant of the transpose equal to the ι-twist of the original determinant. The Tamagawa condition (Tam_V), requiring $H^{1}$(I_q,V) and $H^{1}$(I_q,V^*) to be projective A-modules for primes q away from p, is what makes the local duality maps into isomorphisms and hence makes perfectness possible.","core_discovery":"The central discovery is a functional equation for algebraic Rankin-Selberg p-adic L-functions, stated as char_{HA($Γ_Q^{0}$)} $R^{2}$Γ(V^*(1),D^\\perp) = char_{HA($Γ_Q^{0}$)} $R^{2}$Γ(V,D)^ι. The paper achieves this by identifying the characteristic ideal of the degree-2 Selmer group with the determinant of the Selmer complex, a step that requires the Selmer complex to be perfect with amplitude [1,2] and to have cohomology concentrated in degree 2. These properties are proved for Rankin-Selberg products of Hida families in the ordinary case and of Coleman families in the non-ordinary case, under explicit hypotheses on the Galois representations involved. The main arithmetic consequence is that combined with functional equations for analytic p-adic L-functions, the results strengthen known theorems toward Iwasawa main conjectures by removing the parity conditions that appear in earlier work on symmetric squares and Rankin-Selberg products.","pith_inferences":["This construction suggests that the Tamagawa condition is not merely a technical convenience: if H^1(I_q,V) fails to be projective, the local duality map cannot be an isomorphism, so the determinant step should break and the functional equation should fail in a way that is localizable to the prime q.","The determinant-based method should transfer to other families of automorphic Galois representations, such as those attached to unitary groups or higher symmetric powers, once an analogous control theorem and a Tamagawa-style projectivity statement are proved.","One could test the framework numerically by computing both characteristic ideals at classical specializations for small conductor examples; agreement with the predicted ι-symmetry would support the removal of parity conditions, while a mismatch under the stated hypotheses would expose a flaw in the perfectness argument.","The cylindrical characteristic ideal over HA(Γ_Q^0) may be the natural object for an interpolation formula at mixed-weight classical points; checking such an interpolation is a concrete next step that the paper does not undertake."],"forward_implications":["For Rankin-Selberg products of Hida families, the algebraic p-adic L-function satisfies char = char^ι, so combining with the analytic functional equation removes the parity restriction on the branch of the Iwasawa algebra in earlier main-conjecture results.","For non-ordinary one-parameter Coleman families, the same functional equation holds whenever the Beilinson-Flach Euler system input and the big image hypothesis are available.","For two-parameter Coleman families, the functional equation holds under the Tamagawa projectivity condition, giving multivariate algebraic p-adic L-functions that satisfy the expected symmetric identity.","The determinant-equals-characteristic-ideal theorem provides a uniform definition of algebraic p-adic L-functions over affinoid coefficient rings of arbitrary dimension, not just over the classical Iwasawa algebra.","The same framework also yields a functional equation for algebraic p-adic L-functions attached to adjoint and symmetric-square families once the corresponding perfectness and Tamagawa inputs are verified."],"supporting_citations":[{"why":"Supplies the base perfectness and determinant-equals-characteristic-ideal results for Selmer complexes over ordinary families, and the freeness analysis of H^1(I_v,·) in terms of Tamagawa factors.","marker":"[BCS23]"},{"why":"Provides the Beilinson-Flach Euler system and explicit reciprocity laws used to prove vanishing of degree-1 cohomology and torsion in degree 2.","marker":"[KLZ17]"},{"why":"Supplies the foundational duality formalism for Selmer complexes, including the exact triangle with local error terms that the paper must control.","marker":"[Nek06]"},{"why":"Contains the local and global duality theorems for Selmer complexes over affinoid algebras, including the perfectness criterion for the cup-product pairing.","marker":"[Ben20]"},{"why":"Provides the cohomology theory of arithmetic families of (φ,Γ)-modules used for local duality over HA(Γ^0).","marker":"[KPX14]"},{"why":"Establishes the coadmissible module structure, characteristic ideals, and Iwasawa duality over the rings of tempered distributions used in the punctual case.","marker":"[Pot12]"},{"why":"Supplies the symmetric-square Iwasawa theory bounds that are needed for the functional equation in the symmetric-square case.","marker":"[LZ16a]"},{"why":"Provides the control theorem and perfectness results used to verify the hypotheses for one- and two-parameter Coleman families.","marker":"[Kü23]"}],"fun_headline_variants":["Functional equation proven for Rankin-Selberg p-adic L-functions","Parity conditions removed from Iwasawa main conjectures","Selmer complex determinant yields p-adic L-function functional equation","Rankin-Selberg p-adic L-functions: functional equation without parity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the Tamagawa condition: at every prime away from p, the inertia cohomology $H^{1}$(I_q,V) and $H^{1}$(I_q,V^*) must be free modules over the coefficient ring of the family (equivalently, the p-part of the Tamagawa factor of some classical member must be 1); if this fails, the Selmer complex need not be perfect and the determinant may not equal the characteristic ideal.","fun_headline_variants_meta":{"raw":{"variants":["Functional equation proven for Rankin-Selberg p-adic L-functions","Parity conditions removed from Iwasawa main conjectures","Selmer complex determinant yields p-adic L-function functional equation","Rankin-Selberg p-adic L-functions: functional equation without parity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000515,"raw_usage":{"total_tokens":2484,"prompt_tokens":914,"completion_tokens":1570,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":1493}},"tokens_in":530,"tokens_out":1570,"duration_ms":10837,"temperature":1.0,"reasoning_tokens":1493,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:15:34.387135+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the module $H^{1}$(I_q,V) over the coefficient algebra A = O(X_1) ⊗ O(X_2) for an explicit pair of Coleman families at a prime q where a nontrivial Tamagawa factor is expected; the theorems require this module and its dual to be free, so exhibiting a single torsion element would refute the perfectness statement on which the functional equation rests.","supporting_citations":[],"review_version":1}