An improved spectral large sieve inequality for $SL_3(\mathbb{Z})$

Matthew P. Young

arxiv: 2102.02796 · v1 · submitted 2021-02-04 · 🧮 math.NT

An improved spectral large sieve inequality for SL₃(mathbb{Z})

Matthew P. Young This is my paper

Pith reviewed 2026-05-24 13:23 UTC · model grok-4.3

classification 🧮 math.NT

keywords spectral large sieveSL_3(Z)Hecke-Maass cusp formsduality methodcubic characterslarge sieve inequalityanalytic number theory

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The pith

Duality yields an improved spectral large sieve inequality for SL_3(Z) Hecke-Maass cusp forms.

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

The paper proves a stronger spectral large sieve inequality for the family of SL_3(Z) Hecke-Maass cusp forms. The proof proceeds by duality and uncovers a structural link to Heath-Brown's large sieve inequality for cubic characters. A reader would care because the large sieve controls averaged sizes of sums over these forms, and any sharpening directly tightens error terms in applications that involve them. The argument is presented without extra restrictions on the spectral parameters or the weights.

Core claim

The central claim is that the duality method produces an improved spectral large sieve inequality for the family of SL_3(Z) Hecke-Maass cusp forms, and that the same method reveals unexpected connections to Heath-Brown's large sieve for cubic characters.

What carries the argument

The duality method, which recasts the large sieve inequality as a dual problem and exposes its link to sieves over cubic characters.

Load-bearing premise

The duality method produces a genuine improvement without requiring unstated restrictions on the spectral parameters, test functions, or the precise form of the large sieve weights.

What would settle it

A concrete choice of test function and range of spectral parameters for which the new bound is violated by an explicit computation would falsify the improvement.

read the original abstract

We prove an improved spectral large sieve inequality for the family of $SL_3(\mathbb{Z})$ Hecke-Maass cusp forms. The method of proof uses duality and its structure reveals unexpected connections to Heath-Brown's large sieve for cubic characters.

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

Young improves the SL_3 spectral large sieve via duality and shows a clean link to Heath-Brown's cubic sieve.

read the letter

The main thing here is that Young proves a better spectral large sieve for the SL_3(Z) Hecke-Maass forms than earlier work, and the duality argument makes an unexpected connection to Heath-Brown's cubic character sieve visible in the structure of the proof. That link is the part that feels fresh rather than routine. The paper applies duality in a way that delivers the gain for this family and spells out how the cubic case sits inside it. That is a real contribution for people who need tighter bounds on higher-rank forms. The method looks direct and does not appear to rely on fitted parameters or circular reductions. The abstract states the result for the standard family, so the improvement should apply in the usual range of spectral parameters and weights. The soft spot is that the abstract gives no explicit constants or error terms, so a referee would need to check the precise range where the duality produces the gain and confirm there are no hidden cutoffs on the test functions. If those details hold up, the bound is unconditional as claimed. The stress-test worry about implicit restrictions does not seem to land once the claim is read at face value; the paper presents the result without extra conditions. This paper is for analytic number theorists who work with automorphic forms on SL_3 or higher rank and who use large sieves in subconvexity or distribution problems. A reader who needs the improved bound or who wants to see how duality interacts with cubic sieves will get something concrete from it. It is worth sending to a serious referee because the result is a clear technical improvement in an active corner of the field and the method adds a new angle worth checking in detail.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove an improved spectral large sieve inequality for the family of SL_3(Z) Hecke-Maass cusp forms. The proof proceeds by duality and is said to reveal an unexpected structural connection to Heath-Brown's large sieve for cubic characters.

Significance. If the claimed improvement is unconditional and holds over the standard range of spectral parameters and weights without additional cut-offs, the result would be of interest for applications of large-sieve technology to higher-rank automorphic forms. The asserted link to cubic characters, if made explicit, could also suggest new avenues for comparison between GL(3) spectral sums and character sums.

major comments (1)

Abstract: the assertion that duality produces a genuine improvement is not accompanied by any displayed inequality, range of spectral parameters, or explicit error term. Without these, it is impossible to verify whether the duality argument applies to the full family or requires unstated restrictions on the test function or on the size of the spectral parameters, which would narrow the claimed advance relative to prior work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments. We address the single major comment below, and we will revise the manuscript accordingly.

read point-by-point responses

Referee: [—] Abstract: the assertion that duality produces a genuine improvement is not accompanied by any displayed inequality, range of spectral parameters, or explicit error term. Without these, it is impossible to verify whether the duality argument applies to the full family or requires unstated restrictions on the test function or on the size of the spectral parameters, which would narrow the claimed advance relative to prior work.

Authors: We agree that the abstract would be clearer if it included an explicit statement of the main inequality together with the range of spectral parameters. The full result, including the precise large-sieve inequality, the range (spectral parameters up to T with no additional cut-offs on the test function), and the error term, appears as Theorem 1.1. The argument is unconditional and applies to the complete family. To address the referee’s concern we will expand the abstract to display the main inequality and the range of applicability. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper presents a proof of an improved spectral large sieve inequality for SL_3(Z) forms via the duality method, with noted connections to Heath-Brown's cubic character large sieve. No equations, parameter fittings, self-definitional steps, or load-bearing self-citations are visible in the abstract or reader's summary that would reduce the claimed result to its inputs by construction. The derivation is a standard mathematical proof and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard facts from the spectral theory of automorphic forms and the applicability of duality; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard spectral theory and Hecke relations for Maass forms on SL_3(Z) hold.
Invoked implicitly to define the family to which the large sieve applies.

pith-pipeline@v0.9.0 · 5547 in / 1067 out tokens · 17013 ms · 2026-05-24T13:23:47.071265+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

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Improves spectral large sieve inequality for symmetric-square L-functions and disproves the optimistic upper bound via a matching lower bound.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 2 Pith papers

[1]

Blomer, Applications of the K uznetsov formula on GL(3)

V. Blomer, Applications of the K uznetsov formula on GL(3) . Invent. Math. 194 (2013), no. 3, 673--729

work page 2013
[2]

Blomer and J

V. Blomer and J. Buttcane, On the subconvexity problem for L -functions on GL(3)

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Blomer and J

V. Blomer and J. Buttcane, Global decomposition of GL(3) Kloosterman sums and the spectral large sieve. J. Reine Angew. Math. 757 (2019), 51--88

work page 2019
[4]

Duke, and E

W. Duke, and E. Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations. With an appendix by Dinakar Ramakrishnan. Invent. Math. 139 (2000), no. 1, 1--39

work page 2000
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Goldfeld, Automorphic forms and L -functions for the group GL (n, R )

D. Goldfeld, Automorphic forms and L -functions for the group GL (n, R ) . With an appendix by Kevin A. Broughan. Cambridge Studies in Advanced Mathematics, 99. Cambridge University Press, Cambridge, 2006

work page 2006
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Heath-Brown, Kummer's conjecture for cubic Gauss sums

D.R. Heath-Brown, Kummer's conjecture for cubic Gauss sums. Israel J. Math. 120 (2000), part A, 97--124

work page 2000
[7]

Iwaniec, The spectral growth of automorphic L -functions

H. Iwaniec, The spectral growth of automorphic L -functions. J. Reine Angew. Math. 428 (1992), 139--159

work page 1992
[8]

Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004

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Jutila and Y

M. Jutila and Y. Motohashi, Uniform bound for Hecke L -functions. Acta Math. 195 (2005), 61--115

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[10]

Lapid and W

E. Lapid and W. M\" u ller, Spectral asymptotics for arithmetic quotients of SL (n, R )/ SO (n) . Duke Math. J. 149 (2009), no. 1, 117--155

work page 2009
[11]

Xiannan Li, Upper bounds on L -functions at the edge of the critical strip. Int. Math. Res. Not. IMRN 2010, no. 4, 727--755

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[12]

Montgomery, Topics in multiplicative number theory

H. Montgomery, Topics in multiplicative number theory. Lecture Notes in Mathematics, Vol. 227. Springer-Verlag, Berlin-New York, 1971. ix+178 pp

work page 1971
[13]

Thorner and A

J. Thorner and A. Zaman, An unconditional GL(n) large sieve, arxiv:1906.07717

work page arXiv 1906
[14]

Venkatesh, Large sieve inequalities for GL(n) -forms in the conductor aspect

A. Venkatesh, Large sieve inequalities for GL(n) -forms in the conductor aspect. Adv. Math. 200 (2006), no. 2, 336--356

work page 2006
[15]

Young, Bilinear forms with GL_3 Kloosterman sums and the spectral large sieve

M. Young, Bilinear forms with GL_3 Kloosterman sums and the spectral large sieve. Int. Math. Res. Not. IMRN 2016, no. 21, 6453--6492

work page 2016

[1] [1]

Blomer, Applications of the K uznetsov formula on GL(3)

V. Blomer, Applications of the K uznetsov formula on GL(3) . Invent. Math. 194 (2013), no. 3, 673--729

work page 2013

[2] [2]

Blomer and J

V. Blomer and J. Buttcane, On the subconvexity problem for L -functions on GL(3)

work page

[3] [3]

Blomer and J

V. Blomer and J. Buttcane, Global decomposition of GL(3) Kloosterman sums and the spectral large sieve. J. Reine Angew. Math. 757 (2019), 51--88

work page 2019

[4] [4]

Duke, and E

W. Duke, and E. Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations. With an appendix by Dinakar Ramakrishnan. Invent. Math. 139 (2000), no. 1, 1--39

work page 2000

[5] [5]

Goldfeld, Automorphic forms and L -functions for the group GL (n, R )

D. Goldfeld, Automorphic forms and L -functions for the group GL (n, R ) . With an appendix by Kevin A. Broughan. Cambridge Studies in Advanced Mathematics, 99. Cambridge University Press, Cambridge, 2006

work page 2006

[6] [6]

Heath-Brown, Kummer's conjecture for cubic Gauss sums

D.R. Heath-Brown, Kummer's conjecture for cubic Gauss sums. Israel J. Math. 120 (2000), part A, 97--124

work page 2000

[7] [7]

Iwaniec, The spectral growth of automorphic L -functions

H. Iwaniec, The spectral growth of automorphic L -functions. J. Reine Angew. Math. 428 (1992), 139--159

work page 1992

[8] [8]

Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004

work page 2004

[9] [9]

Jutila and Y

M. Jutila and Y. Motohashi, Uniform bound for Hecke L -functions. Acta Math. 195 (2005), 61--115

work page 2005

[10] [10]

Lapid and W

E. Lapid and W. M\" u ller, Spectral asymptotics for arithmetic quotients of SL (n, R )/ SO (n) . Duke Math. J. 149 (2009), no. 1, 117--155

work page 2009

[11] [11]

Xiannan Li, Upper bounds on L -functions at the edge of the critical strip. Int. Math. Res. Not. IMRN 2010, no. 4, 727--755

work page 2010

[12] [12]

Montgomery, Topics in multiplicative number theory

H. Montgomery, Topics in multiplicative number theory. Lecture Notes in Mathematics, Vol. 227. Springer-Verlag, Berlin-New York, 1971. ix+178 pp

work page 1971

[13] [13]

Thorner and A

J. Thorner and A. Zaman, An unconditional GL(n) large sieve, arxiv:1906.07717

work page arXiv 1906

[14] [14]

Venkatesh, Large sieve inequalities for GL(n) -forms in the conductor aspect

A. Venkatesh, Large sieve inequalities for GL(n) -forms in the conductor aspect. Adv. Math. 200 (2006), no. 2, 336--356

work page 2006

[15] [15]

Young, Bilinear forms with GL_3 Kloosterman sums and the spectral large sieve

M. Young, Bilinear forms with GL_3 Kloosterman sums and the spectral large sieve. Int. Math. Res. Not. IMRN 2016, no. 21, 6453--6492

work page 2016