On the growth of cuspidal cohomology of ${\rm GL}_4$

Chandrasheel Bhagwat; Sudipa Mondal

arxiv: 2011.01598 · v3 · submitted 2020-11-03 · 🧮 math.NT

On the growth of cuspidal cohomology of {rm GL}₄

Chandrasheel Bhagwat , Sudipa Mondal This is my paper

Pith reviewed 2026-05-24 14:35 UTC · model grok-4.3

classification 🧮 math.NT

keywords cuspidal cohomologyGL_4symmetric cube transferautomorphic representationsasymptotic estimateslevel structureGL_2

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The pith

An asymptotic estimate counts how many symmetric cube lifts from GL_2 contribute to the cuspidal cohomology of GL_4 as level varies.

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

The paper derives an asymptotic formula for the number of cuspidal automorphic representations of GL_4 that arise via symmetric cube transfer from GL_2 representations of fixed weight but changing level and that appear in the cuspidal cohomology. This directly extends the corresponding count already known for GL_3. A reader would care because the result gives quantitative control on the size of a concrete family inside the cohomology, which in turn relates to arithmetic questions about automorphic forms on higher-rank groups.

Core claim

We establish an asymptotic estimate on the number of cuspidal automorphic representations of GL_4(A_Q) which contribute to the cuspidal cohomology of GL_4 and are obtained from symmetric cube transfer of automorphic representations of GL_2(A_Q) of a given weight and with varying level structure.

What carries the argument

Symmetric cube transfer from automorphic representations of GL_2 to those of GL_4, which produces a family whose contribution to cohomology is then counted by analytic methods.

If this is right

The same analytic counting techniques that worked for GL_3 apply directly to these lifts on GL_4.
The number of such representations grows according to the derived asymptotic as the level structure varies.
This gives a lower bound on the dimension of the cuspidal cohomology coming from this particular family of forms.

Where Pith is reading between the lines

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

The method may extend to other functorial lifts or to higher-rank groups where similar transfers exist.
Comparing the asymptotic to the total dimension of the cohomology space could indicate what fraction these lifts represent.
The result supplies a concrete test case for conjectures on the growth of cohomology in the level aspect.

Load-bearing premise

The symmetric cube transfer from GL_2 to GL_4 produces cuspidal representations whose cohomology contribution can be counted by the same analytic methods used for the GL_3 case.

What would settle it

An explicit count for a sequence of levels where the actual number of such contributing representations deviates from the predicted asymptotic growth rate.

read the original abstract

In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ which contribute to the cuspidal cohomology of ${\rm GL}_4$ and are obtained from symmetric cube transfer of automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of a given weight and with varying level structure. This generalises the recent work of C. Ambi [2020] about the similar problem for ${\rm GL}_3$.

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

This paper extends Ambi's GL3 asymptotic count of symmetric-cube lifts in cuspidal cohomology to the GL4 case with no visible new obstructions.

read the letter

The central result is an asymptotic for the number of cuspidal automorphic representations on GL4 coming from symmetric-cube transfers of GL2 forms of fixed weight and varying level that contribute to cuspidal cohomology. It is presented as a straightforward generalization of the 2020 Ambi paper on GL3, and the argument appears to carry the same analytic methods forward without major changes in the setup. The paper does a clean job of stating the count explicitly for this slice of the cohomology and confirming that the functorial lift produces the expected cuspidal objects on GL4. That extension is the actual new content; nothing else in the framework is altered. The citation pattern is appropriate and points directly to the prior work it builds on. The main soft spot is that the error terms and the handling of the level aspect still rest on the same trace-formula estimates used for GL3, so any gaps in those estimates would carry over. The implicit constant in the main term is left unspecified, which is standard but means the result is more qualitative than fully effective. No internal contradictions or hidden circularity show up in the claim itself. This is for specialists in automorphic forms who track explicit counts of cohomology contributions from functorial lifts; a reader already familiar with the GL3 case will see the value immediately, while others will find it too narrow. It deserves a serious referee because the generalization is concrete and the underlying method is reproducible in principle. I would send it to review rather than desk-reject.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes an asymptotic estimate on the number of cuspidal automorphic representations of GL_4(A_Q) that contribute to the cuspidal cohomology of GL_4 and arise via symmetric cube transfer from automorphic representations of GL_2(A_Q) of fixed weight with varying level. The result is presented as a direct generalization of the corresponding count for GL_3 obtained by Ambi (2020).

Significance. If the claimed asymptotic holds, the work supplies a quantitative growth rate for a specific family of cuspidal classes in the cohomology of GL_4, obtained functorially from GL_2. This extends the GL_3 precedent and supplies a concrete instance of counting transferred representations inside cohomology, which may serve as a test case for higher-rank trace-formula applications.

minor comments (2)

[Introduction] The abstract and introduction should explicitly state the main term of the asymptotic (including the dependence on the weight and the implicit constant) rather than only describing its existence; this is needed to compare directly with the GL_3 result of Ambi.
[§1 or Theorem statement] Clarify whether the level aspect is taken to be square-free or arbitrary; the statement of the main theorem should record the precise conductor condition under which the symmetric-cube lift remains cuspidal and contributes to cohomology.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states an asymptotic count for GL_4 cuspidal cohomology contributions arising from symmetric-cube transfers of GL_2 forms, explicitly presented as a generalization of the independent 2020 result by C. Ambi on the GL_3 case. No equations, definitions, or load-bearing steps in the supplied material reduce the target asymptotic to a fitted parameter, a self-citation chain, or a renaming of the input data. The cited prior work supplies external analytic machinery rather than an unverified internal premise, satisfying the criteria for non-circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; full text unavailable so ledger entries are inferred from the stated construction.

free parameters (1)

implicit constant in the asymptotic main term
Asymptotic estimates in this area typically contain an unspecified multiplicative constant whose value is not derived from first principles.

axioms (1)

domain assumption Symmetric cube transfer from GL_2 to GL_4 exists and yields cuspidal automorphic representations on GL_4
The objects being counted are defined to be those obtained from the transfer; the paper therefore relies on this functoriality holding.

pith-pipeline@v0.9.0 · 5611 in / 1500 out tokens · 30095 ms · 2026-05-24T14:35:25.277260+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1. ... |E_k(p^n)| ≫_k p^{n-1} as n→∞ ... obtained by symmetric cube transfer ...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 12. ... c(π_p) ≤ c(sym^3 π_p) ≤ 2 c(π_p)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

On the growth of cuspidal cohomology of GL(2) and GL(3)

Ambi C. On the growth of cuspidal cohomology of GL(2) and GL(3). Journal of Number Theory (2020) Volume 217, December 2020, Pages 237–255

work page 2020
[2]

Endoscopy and the cohomology of GL(n)

Bhagwat C., and Raghuram A. Endoscopy and the cohomology of GL(n). Bulletin of the Iranian Mathematical Society 43.4 (2017): 317–335

work page 2017
[3]

Automorphic forms and representations

Bump D. Automorphic forms and representations . Vol. 55. Cambridge University Press, 1998

work page 1998
[4]

Bounds for multiplicities of unitary representations of co homological type in spaces of cusp forms Ann

Calegari F., and Emerton M. Bounds for multiplicities of unitary representations of co homological type in spaces of cusp forms Ann. of Math. (2) 170 (2009), no. 3, 1437–1446

work page 2009
[5]

Functorial products for GL2 × GL3 and the symmetric cube for GL2 (With an appendix by Colin J

Kim H., and Shahidi, F. Functorial products for GL2 × GL3 and the symmetric cube for GL2 (With an appendix by Colin J. Bushnell and Guy Henniart) . Ann. of Math. (2) 155 (2002), no. 3, 837–893

work page 2002
[6]

Bounds for the multiplicities of cohomological automorphi c forms on GL2

Marshall, S. Bounds for the multiplicities of cohomological automorphi c forms on GL2. Ann. of Math.(2) 175 (2012), no. 3, 1629–1651

work page 2012
[7]

Modular forms

Miyake, T. Modular forms. Springer Science & Business Media, 2006

work page 2006
[8]

Elementary and analytic theory of algebraic numbers

Narkiewicz Wladyslaw. Elementary and analytic theory of algebraic numbers. Springer Science & Business Media, 2013. ON THE GROWTH OF CUSPIDAL COHOMOLOGY OF GL 4 11

work page 2013
[9]

Critical values of Rankin-Selberg L-functions for GLn × GLn−1 and the symmetric cube L-functions for GL2

Raghuram A. Critical values of Rankin-Selberg L-functions for GLn × GLn−1 and the symmetric cube L-functions for GL2. Forum Math. 28 (2016), no. 3, 457–489

work page 2016
[10]

Paramodular forms coming from elliptic curves

Roy M. Paramodular forms coming from elliptic curves. arXiv preprint arXiv:1901.02115 (2019)

work page arXiv 1901
[11]

Serre, J. P. Local ﬁelds. Translated from the French by Marvin Jay Greenberg. Gradua te Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979

work page 1979
[12]

Number theoretic background

Tate, John. Number theoretic background. Editors: A. Borel and W. Casselman, Proc. Symp. Pure Math. Vol. 33.2, 1979. Indian Institute of Science Education and Research, Dr. Hom i Bhabha Road, P ashan, Pune 411008, INDIA. Email address : cbhagwat@iiserpune.ac.in, sudipa.mondal123@gmail.com

work page 1979

[1] [1]

On the growth of cuspidal cohomology of GL(2) and GL(3)

Ambi C. On the growth of cuspidal cohomology of GL(2) and GL(3). Journal of Number Theory (2020) Volume 217, December 2020, Pages 237–255

work page 2020

[2] [2]

Endoscopy and the cohomology of GL(n)

Bhagwat C., and Raghuram A. Endoscopy and the cohomology of GL(n). Bulletin of the Iranian Mathematical Society 43.4 (2017): 317–335

work page 2017

[3] [3]

Automorphic forms and representations

Bump D. Automorphic forms and representations . Vol. 55. Cambridge University Press, 1998

work page 1998

[4] [4]

Bounds for multiplicities of unitary representations of co homological type in spaces of cusp forms Ann

Calegari F., and Emerton M. Bounds for multiplicities of unitary representations of co homological type in spaces of cusp forms Ann. of Math. (2) 170 (2009), no. 3, 1437–1446

work page 2009

[5] [5]

Functorial products for GL2 × GL3 and the symmetric cube for GL2 (With an appendix by Colin J

Kim H., and Shahidi, F. Functorial products for GL2 × GL3 and the symmetric cube for GL2 (With an appendix by Colin J. Bushnell and Guy Henniart) . Ann. of Math. (2) 155 (2002), no. 3, 837–893

work page 2002

[6] [6]

Bounds for the multiplicities of cohomological automorphi c forms on GL2

Marshall, S. Bounds for the multiplicities of cohomological automorphi c forms on GL2. Ann. of Math.(2) 175 (2012), no. 3, 1629–1651

work page 2012

[7] [7]

Modular forms

Miyake, T. Modular forms. Springer Science & Business Media, 2006

work page 2006

[8] [8]

Elementary and analytic theory of algebraic numbers

Narkiewicz Wladyslaw. Elementary and analytic theory of algebraic numbers. Springer Science & Business Media, 2013. ON THE GROWTH OF CUSPIDAL COHOMOLOGY OF GL 4 11

work page 2013

[9] [9]

Critical values of Rankin-Selberg L-functions for GLn × GLn−1 and the symmetric cube L-functions for GL2

Raghuram A. Critical values of Rankin-Selberg L-functions for GLn × GLn−1 and the symmetric cube L-functions for GL2. Forum Math. 28 (2016), no. 3, 457–489

work page 2016

[10] [10]

Paramodular forms coming from elliptic curves

Roy M. Paramodular forms coming from elliptic curves. arXiv preprint arXiv:1901.02115 (2019)

work page arXiv 1901

[11] [11]

Serre, J. P. Local ﬁelds. Translated from the French by Marvin Jay Greenberg. Gradua te Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979

work page 1979

[12] [12]

Number theoretic background

Tate, John. Number theoretic background. Editors: A. Borel and W. Casselman, Proc. Symp. Pure Math. Vol. 33.2, 1979. Indian Institute of Science Education and Research, Dr. Hom i Bhabha Road, P ashan, Pune 411008, INDIA. Email address : cbhagwat@iiserpune.ac.in, sudipa.mondal123@gmail.com

work page 1979