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Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

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Pith's one-line read A new class of middle-degree automorphic periods carries a p-adic divisibility between the periods of a GL3(Q) representation and those of its base change to a real quadratic field.

desk verdict A serious, carefully-built paper that defines new middle-degree base-change periods and proves an à la Hida formula for GL3; the advertised divisibility is real but carries an uncomputed local constant u2,ram that keeps the clean headline relation conditional. read the letter →

arxiv 2411.16381 v1 pith:FLB5DIUT submitted 2024-11-25 math.NT

Tristan Ricoul This is my paper · ORCID
classification math.NT MSC 11F6711F7011F7511R4222E55
keywords p-adicperiodsautomorphicbasechangeGL3realquadraticfieldmiddle-degreecohomologycongruencenumbersadjointL-values
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

This paper proves a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its classical base change to a real quadratic field $E$. The new ingredient is a class of periods attached to the middle degree, degree $5$, of the cuspidal cohomology of $\mathrm{GL}_3(E)$, where the $\Pi$-isotypic part is two-dimensional and is split into two one-dimensional eigenspaces by the conjugation or conjugation-duality involution. Under the running hypotheses (Split), (CG), and (LGC$_{m_\pi}$), and with $p$ outside a finite exceptional set, the product $\Omega_2(\pi)\Omega_3(\pi)$ divides $\Omega_5(\Pi,\varepsilon,-)\cdot u$, where $u=u_{2,\mathrm{ram}}$ is an uncomputed local constant. This generalizes the period relations known for $\mathrm{GL}_2$ to $\mathrm{GL}_3$ and gives an automorphic shadow of the conjectural Bloch--Kato period relation for the twisted adjoint motive; a parallel middle-degree divisibility is proved for the stable base change from the unitary group $U_E$.

What carries the argument

The central mechanism is the degree-$5$ cuspidal cohomology $H^5_{\mathrm{cusp}}(Y_E(K_f), L_{\mathfrak{n}}(\mathcal{O}))_{m_\Pi}$, localized at the maximal ideal attached to $\Pi$, viewed as a Hecke module with a semi-linear involution $\iota$ ($\sigma$ for self-conjugate $\Pi$, $\varepsilon$ for conjugate self-dual $\Pi$). The involution splits the two-dimensional $\Pi$-isotypic part into one-dimensional $\pm$-eigenspaces carrying canonical $\mathcal{O}$-structures, which makes possible the canonically normalized Eichler--Shimura maps $\delta^\pm_\iota$ and hence the periods $\Omega_5(\Pi,\iota,\pm)$. The proof of the divisibility uses the congruence-number formalism for Hecke modules with semi-linear involution, the transfer congruence number $\eta^\#_{\lambda_\Pi}(M^*)[\pm]$, and cohomological interpretations of the period integrals that detect classical and stable base changes, with ramified factors computed from essential-vector formulas and archimedean factors from explicit generators of the relevant relative Lie algebra cohomology.

What would settle it

Compute, for an explicit triple $(\pi,E,p)$ satisfying the hypotheses, the $p$-adic valuations of the transfer congruence number $\eta^\#_{\lambda_\Pi}(M^*)[+]$ and of $\Lambda^{\mathrm{imp}}(\pi,\mathrm{Ad}\otimes\chi_E,1)/(\Omega_5(\Pi,\sigma,+)\Omega_5(\Pi^\vee,\sigma,-))$; the claimed divisibility is false if the valuation of the congruence number exceeds that of the quotient. A more direct check of the proof's gate is to test the assumed local-global compatibility at a ramified place above $2$ or above a prime ramified in $E$ for the Galois representation of [CGJ23] when $p$ is completely split in $E$, since the freeness conclusion of Theorem 2.3 is the only bridge from the Hecke algebra to the divisibility.

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Extended reading notes

Core claim

The central claim is Theorem A: if $\pi$ is a self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$, not isomorphic to $\pi\otimes\chi_E$, ramified only at primes of $\mathbb{Q}$ split in $E$, satisfying (Split), (CG), and (LGC$_{m_\pi}$), and if $p$ does not divide $6N_{E/\mathbb{Q}}(\mathfrak{n})h_E(\mathfrak{n})D_E$, then $\Omega_2(\pi)\Omega_3(\pi) \mid \Omega_5(\Pi,\varepsilon,-)\cdot u$, with $u=u_{2,\mathrm{ram}}$ the uncomputed local constant of Theorem C. The underlying discovery is that base-change periods can be defined in middle degree: when $\Pi$ is invariant under the conjugation involution $\sigma$ or the conjugation-duality involution $\varepsilon$, the $\Pi$-isotypic part of degree-$5$ cuspidal cohomology, though two-dimensional, carries a semi-linear involution whose $\pm$-eigenspaces are one-dimensional with canonical integral structure. These middle-degree periods satisfy an adjoint formula of Hida type, Theorem D: up to $p$-adic units, the $\pm$-part of the congruence number of $\Pi$ on $H^5$ equals $\Lambda^{\mathrm{imp}}(\Pi,\mathrm{Ad},1)/(\Omega_5(\Pi,\iota,\pm)\Omega_5(\Pi^\vee,\iota,\mp))$. Combining Theorem D with the factorization of adjoint $L$-functions and of congruence numbers yields the divisibility.

Load-bearing premise

The load-bearing premise is the set of running hypotheses the paper labels (Galm), (LGC), and (CG)—in particular local-global compatibility of the Galois representation attached to the base change at primes where the level ramifies, which the paper notes in Section 2.2.4 is not known when $p$ is completely split in $E$—together with residual absolute irreducibility of $\rho_\Pi$; if any of these fail, the freeness theorem for the localized degree-$5$ cohomology and with it the divisibility collapses.

Editorial extensions

If this is right

  • Under the stated hypotheses, $\Omega_2(\pi)\Omega_3(\pi)$ divides $\Omega_5(\Pi,\varepsilon,-)\cdot u$ for all $p$ avoiding the finite exceptional set, so the classical periods of $\pi$ control the middle-degree period of its base change.
  • The middle-degree periods $\Omega_5(\Pi,\iota,\pm)$ are $p$-integral and intrinsic when $\Pi$ is self-dual, so they can play the role of the automorphic period of the twisted adjoint motive in the Bloch--Kato comparison.
  • The middle-degree adjoint formula makes the normalized imprimitive adjoint value $\Lambda^{\mathrm{imp}}(\Pi,\mathrm{Ad},1)/(\Omega_5(\Pi,\iota,\pm)\Omega_5(\Pi^\vee,\iota,\mp))$ integral, as an element of $\mathcal{O}$.
  • For a self-conjugate stable base change from $U_E$, the twisted adjoint $L$-value of $\pi$ divided by $\Omega_5(\Pi,\sigma,-)$ divides the congruence number of $\pi$ (Corollary 4.1).
  • The transfer divisibility for the stable base change (Theorem 4.1) reduces the missing period relation for $U_E$ to the absence of an adjoint formula of Hida type for that unitary group.

Reading between the lines

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

  • (Inference) If the uncomputed local constant $u_{2,\mathrm{ram}}$ is trivial as expected, Theorem A becomes the clean relation $\Omega_2(\pi)\Omega_3(\pi) \sim \Omega_5(\Pi,\varepsilon,-)$ up to $p$-adic units, and computing that constant in one explicit example would directly test the strength of the theorem.
  • (Inference) The middle-degree construction should carry over to the other cases where the cuspidal range has length two, notably $n=4$ over a real quadratic field, once the archimedean generator computations used in the paper are extended to $\mathrm{GL}_4$; the paper explicitly lists this as Case 2.
  • (Inference) The missing stable-base-change period relation is not caused by the base change itself: the transfer divisibility proved here would upgrade to a full period divisibility once an adjoint formula of Hida type for the unitary group becomes available.
  • (Inference) The one-sided divisibility should be the automorphic shadow of a Bloch--Kato statement: if the reciprocal divisibility can be proved by a non-vanishing-mod-$p$ result for $\mathrm{GL}_3$ of the kind that already underlies the reciprocal divisibility in the $\mathrm{GL}_2$ case, the conjectural equality $\Omega_5(\Pi,\varepsilon,-)\sim\Omega_2(\pi)\Omega_3(\pi)$ would follow.
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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

2 major / 4 minor

Summary. The paper studies p-adic integral period relations for two base-change transfers to GL_3 over a real quadratic field E: the Arthur–Clozel base change from GL_3(Q) and the Rogawski–Mok stable base change from the quasi-split unitary group U_E. Since the middle cohomology of GL_3(E) has two-dimensional isotypic parts, the author introduces new 'base-change periods' Ω_5(Π,ι,±) attached to the ±-eigenspaces of the Galois involution σ (for self-conjugate Π) or the conjugation-duality involution ε (for conjugate self-dual Π). Theorem D (Theorem 3.1) establishes an à la Hida formula: the ±-part of the congruence number η_{λ_Π}(M)[±] equals Λ_imp(Π,Ad,1)/(Ω_5(Π,ι,±)Ω_5(Π∨,ι,∓)) up to O-units. Theorems B and C (Theorems 4.1 and 5.1) then give one-sided divisibilities relating the base-change congruence numbers to twisted adjoint L-values, using a cohomological interpretation of the Flicker–Rallis and Jacquet–Ye periods. The headline Theorem A (Corollary 5.1) deduces the divisibility Ω_2(π)Ω_3(π) | Ω_5(Π,ε,−) up to an uncomputed factor u_{2,ram}. The paper is conditional on the Calegari–Geraghty running hypotheses (Galm), (LGC_m), (Van_m), and (CG), and on residual absolute irreducibility of the associated Galois representation.

Significance. If the core computations are correct, this is a substantive advance: it provides the first middle-degree Hida-style adjoint formula for GL_3 over a real quadratic field and a workable definition of middle-degree periods in a setting where the isotypic cohomology is two-dimensional. The structure is not circular: the periods are fixed by canonically normalized Eichler–Shimura maps and integral O-structures, and the congruence numbers η_{λ_Π}(M)[±] are defined independently of the periods, so Theorem D is a genuine identity rather than a fitted one. The local computations are explicit, using Miyauchi–Matringe essential vectors and Chen's archimedean generators, and the congruence-number formalism is carefully adapted to semi-linear involutions. The main caveat is that the advertised clean period relation (0.0.2) is not actually proved: the theorem contains the uncomputed local factor u_{2,ram}, whose p-adic valuation is not controlled. The theorem is therefore best regarded as a divisibility with a local error term, and the headline claim needs to be rephrased or completed.

major comments (2)
  1. [§5.1 (Theorem C and Corollary 5.1)] The central theorem advertised in the introduction as 0.0.2 is not what Corollary 5.1 proves. The proved relation is Ω_2(π)Ω_3(π) | Ω_5(Π,ε,−)·u_{2,ram}, where u_{2,ram} is an uncomputed nonzero complex factor depending on local components above 2 and at primes ramified in E. Since divisibility is defined after identifying C with Q_p via the fixed isomorphism j_p, the p-adic valuation of u_{2,ram} is load-bearing: if v_p(u_{2,ram})<0, then Ω_5(Π,ε,−)·u_{2,ram} is not integral and the claimed divisibility of periods can fail; if v_p(u_{2,ram})>0, the proved statement is strictly weaker than the clean relation 0.0.2. The text says only that u is 'expected to be trivial', and expectation is not a proof. To make Theorem A a theorem about (0.0.2), the author must either compute u_{2,ram}, prove that it is a p-adic unit, or state the clean divisibility as an explicit conditional consequence.
  2. [§2.2.4 and §2.3.3 (Theorem 2.3 and Theorem A)] Theorem A is conditional on the Calegari–Geraghty hypotheses (Galm), (LGC_{m_π}), and (CG), and the paper itself notes in §2.2.4 that no local-global compatibility at bad places is known for the representation of [CGJ23] when p is completely split in E. Since Theorem A assumes p split in E, the freeness result of Theorem 2.3, and hence the divisibility argument, relies on a conjectural compatibility that is currently open in exactly the relevant case. The hypotheses are declared, so this is not an internal inconsistency, but the abstract and introduction should not present Theorem A as an unconditional proof of a period divisibility; it is a result conditional on those running conjectures. If the local-global compatibility fails, the divisibility argument collapses, and the reader should be told this limitation in the statement of the main theorem as well as in a remark.
minor comments (4)
  1. [§2.1] The differential operator defining the kernel L_n(K) is written as ∂²/(∂X∂A)+∂²/(∂Y∂B)+∂²/(∂X∂A); the third term should presumably be ∂²/(∂Z∂C).
  2. [Abstract and §1.1.2] There are several typos and grammatical slips: 'beetween', 'cus pidal', 'is a ap-adic', and 'when Limp(Π×Π′,s) has not' in §1.2.1. These should be corrected in a final revision.
  3. [Introduction] The introduction contains two unresolved cross-references ('see ?? and ??') when describing the stable base change case; these should be replaced by precise theorem or section numbers.
  4. [§3.1 and §5.1] The notation n is used both for the mirahoric level and for the cohomological weight, which can be confusing; for example, the phrase 'mirahoric level n' and 'cohomological weight n' appear close together in Theorem 3.1. A notational distinction would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the base-change periods and congruence numbers are defined independently before the L-value comparison, and the uncomputed local constant u2,ram is an incompleteness, not a circular step.

full rationale

The central divisibility (Corollary 5.1) is obtained by combining Theorem D (Theorem 3.1), Theorem C (Theorem 5.1), and the factorization L(Π,Ad,s)=L(π,Ad,s)L(π,Ad⊗χE,s). The periods Ω5(Π,ε,±) are defined in §2.4.5 from O-bases of the ±-eigenspaces of the involution ε on the middle-degree cuspidal cohomology, before any L-value is compared; they are not fitted to the divisibility being proved. The congruence numbers ηλ(M)[±] are defined in §2.5.1 as Fitting ideals of M^λ/M_λ, independently of the periods and of the L-values. Theorem D is therefore a genuine identity: the explicit Jacquet–Shalika/Petersson computation and Chen's archimedean calculations determine the normalized pairing, while Lemma 2.5 relates that pairing to the congruence number; period factors cancel because the periods were defined by the same normalized Eichler–Shimura maps, not because the conclusion was assumed. Lemma 2.7 is proved in the text as a general statement about transfer congruence modules with semi-linear involution, and its application to the Flicker–Rallis and Jacquet–Ye periods rests on external vanishing results (Mok, Flicker, Jacquet, Feigon–Lapid–Offen) and on local computations using Miyauchi–Matringe and Chen. The Calegari–Geraghty hypotheses (Galm), (LGCm), (CG) and residual absolute irreducibility are declared assumptions quoted from prior work, not conclusions derived from the target divisibility. The one caveat is the uncomputed local factor u=u2,ram in Theorem C and Corollary 5.1: the paper explicitly states that it is expected to be trivial but hard to compute. If its p-adic valuation is nonzero, the clean divisibility Ω2(π)Ω3(π) | Ω5(Π,ε,−) is not fully quantified. This is a correctness/completeness concern, not a circularity: u is a product of local factors fixed by the local computations and does not depend on the global periods whose divisibility is claimed. No load-bearing self-citation was found; the cited results are either external theorems or prior work with independent content, and the adapted Lemma 2.7 is supplied with a proof. The derivation chain is therefore self-contained: each divisibility is obtained by explicit evaluation of independently defined periods, congruence numbers, and local constants.

Assumptions & free parameters 1 free parameters · 6 assumptions · 1 invented entities

All main results are conditional theorems. The genuinely new objects, the middle-degree periods, are defined canonically and not fitted to the L-values. The load on external conjectures is explicit: (Galm), (LGCm), (Vanm), (CG), plus residual absolute irreducibility of rho_Pi and the self-duality of pi. The product of periods compared with the twisted adjoint L-value is fixed by canonical normalizations up to O^x units, so no parameter was tuned to make the divisibility come out. The uncomputed constant u_{2,ram} is the one place where the statement is not fully explicit.

free parameters (1)
  • u_{2,ram} = uncomputed, expected to be 1
    Local constant in Theorem C depending on the local components of Pi above 2 and at primes ramified in E. It is expected to be trivial but is not computed, so the clean divisibility Omega_2(pi) Omega_3(pi) | Omega_5(Pi,epsilon,-) is not proven.
assumptions (6)
  • domain assumption Conjecture (Galm): a Galois representation rho_m : Gal(Ebar/E) -> GL_n(T) lifting the residual representation and matching Hecke polynomials exists for the localized Hecke algebra T.
    Invoked in Section 2.2.4 and used as a running hypothesis of the Calegari-Geraghty machinery behind Theorem 2.3, which is input to the divisibility proofs.
  • domain assumption Conjecture (LGCm): rho_m satisfies local-global compatibility at minimal, Fontaine-Laflaile, and Taylor-Wiles places.
    Stated in Section 2.2.4; the paper notes no local-global compatibilities at bad places are known for the representation of [CGJ23] when p is completely split in E. Theorem A explicitly assumes (LGC_{m_pi}).
  • domain assumption Conjecture (Vanm): mod p cohomology of YE(Kf) localized at m vanishes outside the Borel-Wallach interval.
    Used to apply the Calegari-Geraghty theorem (Theorem 2.3). The paper states this is proven for n=3, E=Q, but 'remains far from reach' for E real quadratic.
  • domain assumption Assumption (CG): rho_m is S_Kf-minimal, Fontaine-Laflaile at places above p, and residually has enormous image.
    A Calegari-Geraghty running hypothesis, cited in the statement of Theorem A as (CG), making the freeness results and the R=T style arguments conditional.
  • domain assumption Input hypotheses on pi: pi is self-dual, pi is not isomorphic to pi tensor chi_E, and pi satisfies (Split).
    Self-duality guarantees the newform used to define the opposite base-change periods is a good test vector for the Jacquet-Ye or Flicker-Rallis period, as explained in the introduction's discussion of test vectors. Without these hypotheses the divisibility statements are not claimed.
  • domain assumption Residual absolute irreducibility of the Galois representation rho_Pi associated with the base change Pi.
    Required for uniqueness of the O-lattice of rho_Pi and for Lemma 4.1 showing the split-part Hecke algebra coincides with the full localized Hecke algebra. The paper cannot currently verify this assumption in examples.
invented entities (1)
  • Middle-degree base-change periods Omega_5(Pi,sigma,pm) and Omega_5(Pi,epsilon,pm) independent evidence
    purpose: 1-dimensional integral periods cut out of the 2-dimensional middle cuspidal cohomology H^5 by the Galois involution sigma or the conjugate-dual involution epsilon, replacing the extremal top and bottom degree periods.
    The periods are canonically defined via normalized Eichler-Shimura maps and integral O-structures, and they satisfy the a la Hida formula of Theorem D and appear in the divisibility theorems, so they are testable mathematical objects rather than ad hoc postulates.

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Pith. "Pith review of Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations." pith.science (2026). https://pith.science/paper/FLB5DIUT

@misc{pith2026241116381,
  author       = {Pith},
  title        = {Pith review of: Real quadratic base changes for $\mathrmGL_3$ and integral periods relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLB5DIUT}},
  note         = {Machine review of arXiv:2411.16381}
}
abstract

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

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