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Functional equations of algebraic Rankin-Selberg $p$-adic $L$-functions

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Pith's one-line read 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…

desk verdict Solid determinant-based functional equations, but the advertised parity removal in the non-ordinary 2-parameter case rests on an unproved Tamagawa projectivity hypothesis. read the letter →

arxiv 2412.11147 v1 pith:YXTZDYPJ submitted 2024-12-15 math.NT

Kâz{i}m Büyükboduk , Manisha Ganguly This is my paper · ORCID
classification math.NT MSC 11R23
keywords IwasawatheoryRankin-Selbergproductsp-adicL-functionsfunctionalequationsSelmercomplexesTamagawafactorsColemanfamiliesHida
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

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.

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 (5)
  1. [§6.3.2, Theorem 6.20] 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.
  2. [§6.3.2, Theorem 6.19] 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.
  3. [§5.7–5.8, Proposition 5.22, Theorems 5.24 and 5.25] 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.
  4. [§5.6.6, Proposition 5.17] 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).
  5. [§6.4, Theorem 6.23] 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.
minor comments (5)
  1. [§2.5] The sentence 'In this subsection, where we closely follow closely follow [LZ16a]' contains a duplicated phrase 'closely follow closely follow'; it should be corrected.
  2. [Assumption 2.7(3)] 'Hece field' should be 'Hecke field'.
  3. [§2.1.2] 'of of Krull-dimension 3' contains a duplicated 'of'.
  4. [§6.3.2, Theorem 6.19 proof] 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).
  5. [§6.3.1, Proposition 6.14] 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.

Circularity Check

2 steps flagged · score 2.0 of 10

No substantive circularity: the functional equation is deduced from external duality and perfectness results, with the non-ordinary 2-parameter case explicitly conditional on (TamV); two self-referential proof typos are flagged but are not load-bearing.

  1. other [Theorem 6.23, proof (Section 6.4)]
    "This is a special case of Theorem 5.25, where we note that all the assumptions required in Theorem 5.25 hold in this scenario thanks to Theorem 6.23."

    Taken literally, the proof of Theorem 6.23 invokes Theorem 6.23 itself to verify the hypotheses of Theorem 5.25, which would be circular. The evident intended reference is Theorem 6.22, a result of Kuecuek that is independent of the present paper's conclusion. The self-reference is therefore a typographical error rather than a substantive circular step, and it does not support the paper's main functional equations.

  2. other [Theorem 6.20, proof (Section 6.3.2)]
    "det HA(Γ0Q) RΓ(V∗(1),D⊥) = det HA(Γ0Q) RΓ(V∗(1),D⊥) = det HA(Γ0Q) RΓ(V,D)ι = det HA(Γ0Q) RΓ(V,D)ι."

    The displayed chain begins and ends with identical expressions, so the first and last equalities are tautological and carry no content. The intended nontrivial equality is the middle one, coming from Selmer-complex duality. This is a proof-writing typo, not a reduction of the theorem to its own statement, and the surrounding argument still relies on Theorem 5.18 and the det = char comparison.

full rationale

The derivation chain is essentially self-contained and external. In Part 2, the abstract functional equation (Theorem 5.25) follows from Selmer-complex duality (Theorem 5.18, attributed to Benois) plus the determinant-characteristic-ideal comparison (Theorem 5.24), which in turn uses Proposition 3.5 (an algebra fact for coadmissible modules over HA(Γ0F)) and Proposition 5.22 (perfect amplitude, sourced to Kuecuek). The proof of Proposition 3.5 cites [BCS23, Proposition 4.44] for a local DVR statement, but that is auxiliary algebra, not the target functional equation. In the arithmetic applications, the ordinary case (Theorem 2.15) uses [BCS23] for Tamagawa freeness and [KLZ17] for Beilinson-Flach Euler systems to control the rank; these are independent inputs that do not contain the conclusion. The 1-parameter non-ordinary case (Theorem 6.16) uses Kuecuek's perfectness theorem and the projectivity of H^1(I_q,W). The 2-parameter non-ordinary case (Theorem 6.20) is explicitly conditional on (TamV), namely projectivity of H^1(I_q,V) and H^1(I_q,V*); this is a stated hypothesis, not a disguised form of the conclusion, so the conditional nature of the advertised parity removal is a limitation or open verification, not circularity. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors is invoked to forbid alternatives, and no ansatz is smuggled in via citation. The two flagged passages are proof typos: Theorem 6.23 cites itself (clearly meant Theorem 6.22), and Theorem 6.20 contains a tautological first equality. Both are localized and non-load-bearing. Overall circularity is minimal: score 2.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical parameters are fitted; the assumptions above are the input hypotheses on Galois representations and local conditions. No new physical or algebraic entity is postulated; the cylindrical characteristic ideal is a defined object, not an independent entity.

assumptions (5)
  • domain assumption Residual representations rho_tilde_f and rho_tilde_g are absolutely irreducible, and the pair satisfies (Conj), (nCM), (nTriv), (Reg), (nZ), (Nob), and p >= 7 where required.
    These assumptions appear in Section 1.3.3 and are used to ensure irreducibility of tensor products (Theorem A.2) and validity of the control theorem for Selmer complexes.
  • domain assumption Tamagawa condition: for q in Sigma_p, H^1(I_q,V) and H^1(I_q,V*) are projective A-modules, equivalently the p-part of the Tamagawa factor at v not dividing p equals 1 for some member of each family.
    This is the load-bearing hypothesis that makes the Selmer complexes perfect and the duality map an isomorphism; it is not proved for all inputs and enters at Proposition 2.9 and Theorem 6.19.
  • standard math Beilinson-Flach Euler system exists and the non-vanishing L(f,g,1+j) != 0 holds.
    Used in Theorem 6.9 to force vanishing of R^1 and torsionness of R^2 for punctual Selmer complexes; the paper cites Kings-Loeffler-Zerbes and Loeffler-Zerbes.
  • domain assumption Big Image Hypothesis (BISym) for symmetric-square applications.
    Imported from Loeffler-Zerbes [LZ16a] to bound Selmer groups of symmetric square representations in Theorem 2.18 and used in the adjoint application.
  • standard math Theorem 6.22 of Kucuk and his forthcoming addendum relaxing condition (c) after Remark 7.2 are valid.
    The paper uses these results to prove perfectness and vanishing in the adjoint case; the addendum is not available at the time of writing.

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Pith. "Pith review of Functional equations of algebraic Rankin-Selberg $p$-adic $L$-functions." pith.science (2026). https://pith.science/paper/YXTZDYPJ

@misc{pith2026241211147,
  author       = {Pith},
  title        = {Pith review of: Functional equations of algebraic Rankin-Selberg $p$-adic $L$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YXTZDYPJ}},
  note         = {Machine review of arXiv:2412.11147}
}
abstract

This article presents an approach to the algebraic functional equation for Selmer complexes, which in turn have applications in the Iwasawa theoretic study of Rankin-Selberg products of the Hida and Coleman families. Our treatment establishes the functional equation for algebraic $p$-adic $L$-functions (which are given in terms of characteristic ideals of Selmer groups, which arise as the cohomology of appropriately defined Selmer complexes in degree $2$). This is achieved by recovering the characteristic ideal as the determinant of the said Selmer complex, once we prove (under suitable but rather mild) hypotheses that the Selmer complex in question is perfect with amplitude $[1,2]$, and its cohomology is concentrated in degree-2. The perfectness of these Selmer complexes turns out to be a delicate problem, and the required properties require a study of Tamagawa factors in families, which may be of independent interest.

Discussion (0). Continue with ORCID to comment.

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    2014

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.