Critical values of $L$-functions of residual representations of $\mathrm{GL}_4$

Johannes Droschl

arxiv: 2407.13464 · v3 · submitted 2024-07-18 · 🧮 math.NT

Critical values of L-functions of residual representations of GL₄

Johannes Droschl This is my paper

Pith reviewed 2026-05-23 22:57 UTC · model grok-4.3

classification 🧮 math.NT

keywords L-functionsresidual spectrumGL(4)Jacquet-Langlands correspondenceShalika periodscritical valuesautomorphic representationsrationality

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The pith

Critical values of L-functions for residual representations of GL_4 equal Shalika periods times a rational multiple.

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

The paper proves rationality results for critical values of L-functions attached to representations in the residual spectrum of GL_4 over the adeles. It applies the Jacquet-Langlands correspondence to transfer the partial L-functions to those of cuspidal automorphic representations on the inner form GL_2' over a quaternion algebra. Building on ideas from Grobner and Raghuram, the critical values are then identified with Shalika periods up to a rational factor. A reader would care because this makes the arithmetic content of these special values explicit in terms of periods that can be studied independently.

Core claim

We prove rationality results of critical values for L-functions attached to representations in the residual spectrum of GL_4(A). We use the Jacquet-Langlands correspondence to describe their partial L-functions via cuspidal automorphic representations of the group GL_2'(A) over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.

What carries the argument

Jacquet-Langlands correspondence transferring residual GL_4 representations to cuspidal ones on the quaternion algebra, followed by extraction of Shalika periods from the transferred L-functions.

If this is right

The rationality of the critical L-values follows directly once the rationality of the Shalika periods is known.
The partial L-functions of these residual representations coincide with those arising from cuspidal representations on the quaternion algebra.
The Grobner-Raghuram period extraction applies to this class of residual representations.
Rational multiples relating L-values and periods are determined by the transfer.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same transfer-plus-period method may apply to residual spectra of GL_n for n>4 when suitable inner forms exist.
Explicit computations of Shalika periods for low-discriminant quaternion algebras could yield numerical checks of the rationality statements.
The results suggest that residual-spectrum L-values fit into the same period conjectures already formulated for cuspidal representations.

Load-bearing premise

The Jacquet-Langlands correspondence applies to residual representations of GL_4 so that partial L-functions match those of cuspidal representations on the quaternion algebra side, and the method for extracting Shalika periods extends without additional obstructions.

What would settle it

An explicit residual representation of GL_4(A) for which the ratio of a critical L-value to the corresponding Shalika period is irrational.

read the original abstract

In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial $L$-functions via cuspidal automorphic representations of the group $\mathrm{GL}_2'(\mathbb{A})$ over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Grobner-Raghuram rationality results to residual GL4 L-functions by JL transfer to cuspidal reps on a quaternion algebra, but the transfer step for residuals is assumed without visible checks and could introduce non-rational factors.

read the letter

The main takeaway is that this work applies the Jacquet-Langlands correspondence to move residual spectrum representations of GL4 to cuspidal ones on a quaternion algebra inner form, then uses Grobner-Raghuram style arguments to express the critical L-values in terms of Shalika periods up to rationals. What stands out as new is the focus on the residual spectrum case for GL4, which hasn't been treated this way before according to the abstract. It builds directly on established tools without claiming a new general method. The paper does well in clearly outlining the strategy in the abstract: transfer the L-functions, then extract the period. This keeps the argument tied to existing results on periods. The soft spot is the assumption that JL applies directly to residual representations while preserving the partial L-functions at critical points. Residual representations arise as residues of Eisenstein series from GL2 x GL2 data, and JL correspondences are typically formulated for cuspidal or discrete spectrum. Any mismatch in how the correspondence handles the constant terms or normalizing factors could introduce non-rational multipliers into the critical value. The abstract invokes the correspondence without mentioning a separate verification for the residual case, so this step carries the main risk. The citation pattern looks standard, relying on Grobner-Raghuram and JL literature. No obvious circularity since it reduces to those priors. This paper is aimed at researchers working on automorphic L-functions, periods, and inner forms of GLn. Someone already versed in the Grobner-Raghuram method would find it a useful case study, even if they have to fill in the details themselves. It deserves peer review because the claim is concrete and the method is traceable, though the referee would need to focus on whether the JL application holds for residuals without extra factors. I would send it out rather than desk reject.

Referee Report

1 major / 1 minor

Summary. The manuscript proves rationality results for the critical values of L-functions attached to representations in the residual spectrum of GL_4(A). It invokes the Jacquet-Langlands correspondence to equate the partial L-functions of these residual representations with those of cuspidal representations on the inner form GL_2'(A) over a quaternion algebra, then applies techniques inspired by Grobner and Raghuram to express the critical values in terms of Shalika periods up to a rational multiple.

Significance. If the central claims hold, the work extends rationality results for critical L-values from the cuspidal to the residual spectrum of GL_4, a meaningful contribution to the study of special values in the Langlands program. The explicit reduction to Shalika periods supplies a concrete computational mechanism. The manuscript appropriately credits the Grobner-Raghuram framework as the source of the extraction method.

major comments (1)

[Section describing the method] The section applying the Jacquet-Langlands correspondence (the paragraph describing the method and any preceding setup of the residual spectrum): the claim that partial L-functions of residual representations of GL_4 match those of cuspidal representations on the quaternion algebra side, preserving critical points and local factors exactly, is load-bearing for the rationality statement. Standard statements of JL for inner forms of GL_4 are formulated for the cuspidal or discrete spectrum; residual representations arise as residues of Eisenstein series (typically from GL_2 × GL_2 data), and any mismatch in constant terms or normalizing factors would introduce non-rational multipliers. Explicit verification that the correspondence maps the residual spectrum into the cuspidal spectrum on the inner form without altering the critical values is required.

minor comments (1)

[Introduction] The abstract is concise, but the introduction would benefit from an explicit statement of the main theorem, including the precise hypotheses on the residual representations (e.g., the inducing data and the places where the representation is residual).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this key point concerning the application of the Jacquet-Langlands correspondence. We address the comment below and will revise the manuscript to incorporate the requested verification.

read point-by-point responses

Referee: [Section describing the method] The section applying the Jacquet-Langlands correspondence (the paragraph describing the method and any preceding setup of the residual spectrum): the claim that partial L-functions of residual representations of GL_4 match those of cuspidal representations on the quaternion algebra side, preserving critical points and local factors exactly, is load-bearing for the rationality statement. Standard statements of JL for inner forms of GL_4 are formulated for the cuspidal or discrete spectrum; residual representations arise as residues of Eisenstein series (typically from GL_2 × GL_2 data), and any mismatch in constant terms or normalizing factors would introduce non-rational multipliers. Explicit verification that the correspondence maps the residual spectrum into the cuspidal spectrum on the inner form without altering the critical values is required.

Authors: We agree that standard statements of the Jacquet-Langlands correspondence are formulated for cuspidal representations and that residual representations on GL_4 arise as residues of Eisenstein series from GL_2 × GL_2 data. In the manuscript the transfer is effected by applying the correspondence to the cuspidal data on the quaternion algebra side that underlies the inducing data. To ensure the partial L-functions coincide exactly (including at critical points) and that no non-rational factors arise from constant terms or normalizing factors, we will add an explicit verification in the revised version. This will consist of a direct comparison of the local factors at all places together with a check that the global L-function definitions and the residue construction are compatible under the correspondence, thereby confirming that the critical values are preserved up to the rational multiple already stated. revision: yes

Circularity Check

0 steps flagged

No circularity; results rest on external correspondences and prior theorems.

full rationale

The derivation invokes the Jacquet-Langlands correspondence to equate partial L-functions of residual GL_4 representations with those of cuspidal representations on the inner form, then adapts the Grobner-Raghuram extraction of Shalika periods to obtain rationality up to Q^×. These steps cite independent prior results (not self-citations or internal fits) whose validity is external to the paper. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citation chains appear; the central claim remains non-tautological against the stated external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger constructed from abstract only; full paper would list additional background results on L-functions and periods.

axioms (2)

domain assumption Jacquet-Langlands correspondence preserves the partial L-functions attached to the relevant representations.
Invoked to transfer the L-functions from GL_4 residual spectrum to the quaternion algebra side.
domain assumption The method of Grobner and Raghuram for expressing critical values as Shalika periods applies directly in this setting.
Used to obtain the rationality statement.

pith-pipeline@v0.9.0 · 5610 in / 1362 out tokens · 25599 ms · 2026-05-23T22:57:38.721066+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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D. A. Vogan, Jr. and G. J. Zuckerman. Unitary representations with nonzero cohomology. Compositio Math., 53(1):51–90, 1984

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J.-L. Waldspurger. Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2). Compositio Math., 54(2):121–171, 1985

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A. V. Zelevinsky. Induced representations of reductivep-adic groups. II. On irreducible representations ofGL(n). Annales scientifiques de l’École Normale Supérieure, 13(2):165–210, 1980. 66

work page 1980

[1] [1]

A. Ash. Non-square-integrable cohomology of arithmetic groups.Duke Math. J., 47(2):435–449, 1980

work page 1980

[2] [2]

A. I. Badulescu. Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations.Inventiones mathematicae, 172(2):383–438, Jan 2008

work page 2008

[3] [3]

A. I. Badulescu and D. Renard. Unitary dual of GL(n) at archimedean places and global Jacquet–Langlands correspondence.Compositio Mathematica, 146(5):1115–1164, Jun 2010

work page 2010

[4] [4]

A. Borel. Some finiteness properties of adele groups over number fields.Publi- cations Mathématiques de l’IHÉS, 16:5–30, 1963. 64

work page 1963

[5] [5]

Borel and N

A. Borel and N. R. Wallach.Continuous cohomology, discrete subgroups, and representations of reductive groups, volume No. 94 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1980

work page 1980

[6] [6]

C. J. Bushnell and A. Fröhlich.Gauss sums andp-adic division algebras, volume 987 ofLecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1983

work page 1983

[7] [7]

L. Clozel. Motifs et formes automorphes: Applications du principe de fonctori- alité. Automorphic forms, Shimura varieties, and L-functions, Vol. I, Perspect. Math., vol. 10, eds. L. Clozel and J. S. Milne:77–159, 1988

work page 1988

[8] [8]

Deligne, P

D. Deligne, P. Kazhdan and M.-F. Vignéras. Représentations des algèbres centrales simplesp-adiques. Représentations des groupes réductifs sur un corps local, Travaux en Cours, 33-117 (1984)., 1984

work page 1984

[9] [9]

Friedberg and H

S. Friedberg and H. Jacquet. Linear periods.J. Reine Angew. Math., 443:91–139, 1993

work page 1993

[10] [10]

W. T. Gan and S. Takeda. On Shalika periods and a theorem of Jacquet-Martin. Amer. J. Math., 132(2):475–528, 2010

work page 2010

[11] [11]

Godement

R. Godement. Notes on Jacquet-Langlands’ theory, volume 8 ofCTM. Clas- sical Topics in Mathematics. Higher Education Press, Beijing, 2018. With commentaries by Robert Langlands and Herve Jacquet

work page 2018

[12] [12]

Godement and H

R. Godement and H. Jacquet.Zeta functions of simple algebras, volume Vol. 260 ofLecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1972

work page 1972

[13] [13]

H. Grobner. Smooth-automorphic forms and smooth-automorphic represen- tations, volume 17 ofSeries on Number Theory and its Applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, [2023]©2023

work page 2023

[14] [14]

Grobner and A

H. Grobner and A. Raghuram. On some arithmetic properties of automorphic forms of GLm over a division algebra.Int. J. Number Theory, 10(4):963–1013, 2014

work page 2014

[15] [15]

Grobner and A

H. Grobner and A. Raghuram. On the arithmetic of Shalika models and the critical values ofL-functions for GL2n. Amer. J. Math., 136(3):675–728, 2014. With an appendix by Wee Teck Gan

work page 2014

[16] [16]

Jacquet and J

H. Jacquet and J. Shalika. Exterior squareL-functions. In Automorphic forms, Shimura varieties, andL-functions, Vol. II (Ann Arbor, MI, 1988), volume 11 of Perspect. Math., pages 143–226. Academic Press, Boston, MA, 1990

work page 1988

[17] [17]

Jiang, B

D. Jiang, B. Sun, and F. Tian. Period Relations for StandardL-functions of Symplectic Type. arXiv e-prints, page arXiv:1909.03476, Sept. 2019

work page arXiv 1909

[18] [18]

R. P. Langlands. On the notion of an automorphic representation. a supplement to the preceding paper.Proc. Sympos. Pure Math, 33(part I):203–207, 1979

work page 1979

[19] [19]

LocalRankin-SelbergintegralsforSpehrepresentations

E.M.LapidandZ.Mao. LocalRankin-SelbergintegralsforSpehrepresentations. Compos. Math., 156(5):908–945, 2020. 65

work page 2020

[20] [20]

Mœglin and J

C. Mœglin and J. L. Waldspurger. Le spectre résiduel deGL(n). Ann. Sci. École Norm. Sup. (4), 22(4):605–674, 1989

work page 1989

[21] [21]

C. Nien. Uniqueness of Shalika models.Canad. J. Math., 61(6):1325–1340, 2009

work page 2009

[22] [22]

Platonov and A

V. Platonov and A. Rapinchuk.Algebraic groups and number theory, volume 139 ofPure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen

work page 1994

[23] [23]

Raghuram and F

A. Raghuram and F. Shahidi. On certain period relations for cusp forms on GLn. Int. Math. Res. Not. IMRN, pages Art. ID rnn 077, 23, 2008

work page 2008

[24] [24]

S. A. Salamanca-Riba. On the unitary dual of real reductive Lie groups and the Ag(λ) modules: the strongly regular case.Duke Math. J., 96(3):521–546, 1999

work page 1999

[25] [25]

T. A. Springer.Linear algebraic groups. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, second edition, 2009

work page 2009

[26] [26]

B. Sun. Cohomologically induced distinguished representations and cohomolog- ical test vectors.Duke Math. J., 168(1):85–126, 2019

work page 2019

[27] [27]

D. A. Vogan, Jr. and G. J. Zuckerman. Unitary representations with nonzero cohomology. Compositio Math., 53(1):51–90, 1984

work page 1984

[28] [28]

Waldspurger

J.-L. Waldspurger. Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2). Compositio Math., 54(2):121–171, 1985

work page 1985

[29] [29]

Weil.Basic number theory, volume Band 144 ofDie Grundlehren der math- ematischen Wissenschaften

A. Weil.Basic number theory, volume Band 144 ofDie Grundlehren der math- ematischen Wissenschaften. Springer-Verlag, New York-Berlin, third edition, 1974

work page 1974

[30] [30]

A. V. Zelevinsky. Induced representations of reductivep-adic groups. II. On irreducible representations ofGL(n). Annales scientifiques de l’École Normale Supérieure, 13(2):165–210, 1980. 66

work page 1980