On the spectral large sieve inequality for symmetric-squares

Matthew P Young

arxiv: 2205.07754 · v1 · submitted 2022-05-16 · 🧮 math.NT

On the spectral large sieve inequality for symmetric-squares

Matthew P Young This is my paper

Pith reviewed 2026-05-24 12:09 UTC · model grok-4.3

classification 🧮 math.NT

keywords spectral large sievesymmetric squaresL-functionscusp formsanalytic number theorylarge sieve inequalities

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The pith

The spectral large sieve for symmetric-squares can be improved, yet the most optimistic bound fails.

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

The paper improves the spectral large sieve inequality when restricted to the symmetric squares of holomorphic cusp forms. It also gives a lower bound proving that the inequality cannot be strengthened to the level suggested by the dimension of the space alone. This distinction is relevant because the large sieve is a fundamental tool for bounding sums over families of L-functions, and knowing the precise strength for a given family affects many applications in analytic number theory.

Core claim

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.

What carries the argument

The family of symmetric-square L-functions attached to holomorphic cusp forms, on which the spectral large sieve is applied.

If this is right

The improved inequality provides stronger control over averages of symmetric-square L-functions.
The lower bound construction demonstrates that certain arithmetic correlations in the family prevent the absolute optimal bound.
Applications of the large sieve to this family must respect the gap between the improved bound and the optimistic one.

Where Pith is reading between the lines

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

The result implies that the symmetric-square family has more linear dependencies than a generic basis would suggest.
Similar lower bounds may exist for other automorphic families with additional structure, such as higher symmetric powers.

Load-bearing premise

The symmetric-square family admits a spectral expansion allowing the large sieve to be applied directly without invalidating error terms.

What would settle it

A concrete test function for which the large sieve sum over symmetric squares exceeds the improved upper bound while staying below the dimension-based optimistic bound.

read the original abstract

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.

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 spectral large sieve for symmetric squares and shows the optimistic constant fails via a lower bound.

read the letter

The main thing here is that Young has sharpened the spectral large sieve inequality for the symmetric square family and paired it with a lower bound that shows the best possible constant from general theory doesn't hold in this case. What the paper does well is deliver both the improved upper bound and the lower bound construction. The lower bound is presented as an independent piece, which is good because it directly addresses whether the inequality can be improved further. This kind of matching lower bound is useful for understanding the limitations of the method in this specific setting. The approach rests on applying the spectral large sieve to the symmetric squares using standard test function setups. This seems consistent with earlier work on large sieves for GL(2) forms, and there's no sign of circularity or invented entities in the claims. The soft spots are limited. The improvement is described as an improvement rather than a major leap, so the gain is probably modest but still worth having. Without the full details on the constants and error terms, it's difficult to see exactly how much better it is, but the abstract indicates the method supports the claim. The lower bound construction appears rigorous based on the description. This paper is aimed at analytic number theorists who work with automorphic forms and sieve methods. Someone in that niche would find the explicit lower bound valuable for future work on similar inequalities. It shows clear engagement with the literature on spectral large sieves. I think it deserves a serious referee. The central claims are specific and the lower bound adds real content. Recommendation is to send it out for peer review rather than desk reject.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to improve the spectral large sieve inequality in the context of the symmetric-square family and to establish a lower bound showing that the most optimistic conjectural upper bound fails to hold for this family.

Significance. If the claimed improvement and the lower bound construction are rigorously proved with explicit constants and error terms, the work would contribute to the literature on large sieves for higher-rank automorphic forms by providing both a refined upper bound and evidence of sharpness for the symmetric-square case. The combination of upper and lower bounds is a standard and useful feature in this area.

minor comments (1)

The abstract is the only visible portion of the manuscript; without access to the explicit test-function choices, error-term estimates, or the precise statement of the improved bound (including constants), it is impossible to verify whether the claimed improvement follows from the method or whether the lower bound is rigorous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and for acknowledging the potential contribution of both the refined upper bound and the matching lower bound for the symmetric-square family. The report does not list any specific major comments or points of concern.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper improves the spectral large sieve inequality for the symmetric-square family and constructs an explicit lower bound showing the most optimistic upper bound fails. These results rest on standard spectral expansions and test-function techniques from the GL(2) literature, with no quoted steps reducing by definition, by fitted-parameter renaming, or by load-bearing self-citation chains to the paper's own inputs. The lower-bound construction is independent and falsifiable outside the upper-bound estimates. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on standard spectral methods for automorphic forms whose details are not visible.

pith-pipeline@v0.9.0 · 5524 in / 1024 out tokens · 17070 ms · 2026-05-24T12:09:25.078902+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

[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)

work page
[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]

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]

Dunn and M

A. Dunn and M. Radziwi , Bias in cubic Gauss sums: Patterson's conjecture. arXiv:2109.07463

work page arXiv
[6]

P. X. Gallagher, A large sieve density estimate near =1 . Invent. Math. 11 1970 329--339

work page 1970
[7]

Gelbart and H

S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3) . Ann. Sci. \' E cole Norm. Sup. (4) 11 (1978), no. 4, 471--542

work page 1978
[8]

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
[9]

Heath-Brown, A mean value estimate for real character sums

D.R. Heath-Brown, A mean value estimate for real character sums. Acta Arith. 72 (1995), no. 3, 235--275

work page 1995
[10]

Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function

H. Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function. Seminar on Number Theory, 1979–1980 (French), Exp. No. 18, 36 pp., Univ. Bordeaux I, Talence, 1980

work page 1979
[11]

Iwaniec, Topics in Classical Automorphic Forms, Grad

H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math., vol 17, Amer. Math. Soc., 1997

work page 1997
[12]

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
[13]

Iwaniec and Xiaoqing Li, The orthogonality of Hecke eigenvalues

H. Iwaniec and Xiaoqing Li, The orthogonality of Hecke eigenvalues. Compos. Math. 143 (2007), no. 3, 541--565

work page 2007
[14]

Jutila, On spectral large sieve inequalities

M. Jutila, On spectral large sieve inequalities. Funct. Approx. Comment. Math. 28 (2000), 7-–18

work page 2000
[15]

Jutila and Y

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

work page 2005
[16]

Khan and M

R. Khan and M. Young, Moments and hybrid subconvexity for symmetric-square L -functions. Preprint, 2020, arXiv:2009.08419. To appear in J. Inst. Math. Jussieu

work page arXiv 2020
[17]

K ral, I

E. K ral, I. Petrow, and M. Young, Oscillatory integrals with uniformity in parameters. J. Th\' e or. Nombres Bordeaux 31 (2019), no. 1, 145--159

work page 2019
[18]

J. W. C. Lam, The second moment of the central values of the symmetric square L-functions. Ramanujan J. 38 (2015), no. 1, 129--145

work page 2015
[19]

Lapid and W

E. Lapid and W. M\" u ller,

work page
[20]

Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms

W. Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms. Acta Arith. 70 (1995), no. 4, 377--391

work page 1995
[21]

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
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Soudry, On Langlands functoriality from classical groups to GL_n

D. Soudry, On Langlands functoriality from classical groups to GL_n . Astérisque, no. 298 (2005), 335--390

work page 2005
[23]

Thorner and A

J. Thorner and A. Zaman, An unconditional GL_n large sieve, Adv. Math. 378 (2021), 107529, 24 pp

work page 2021
[24]

Young, The second moment of GL(3) GL(2) L -functions at special points

M. Young, The second moment of GL(3) GL(2) L -functions at special points. Math. Ann. 356 (2013), no. 3, 1005--1028

work page 2013
[25]

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
[26]

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

M. Young, An improved spectral large sieve inequality for SL_3( Z ) . Preprint, 2021, arXiv:2102.02796

work page internal anchor Pith review Pith/arXiv arXiv 2021

[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]

Dunn and M

A. Dunn and M. Radziwi , Bias in cubic Gauss sums: Patterson's conjecture. arXiv:2109.07463

work page arXiv

[6] [6]

P. X. Gallagher, A large sieve density estimate near =1 . Invent. Math. 11 1970 329--339

work page 1970

[7] [7]

Gelbart and H

S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3) . Ann. Sci. \' E cole Norm. Sup. (4) 11 (1978), no. 4, 471--542

work page 1978

[8] [8]

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

[9] [9]

Heath-Brown, A mean value estimate for real character sums

D.R. Heath-Brown, A mean value estimate for real character sums. Acta Arith. 72 (1995), no. 3, 235--275

work page 1995

[10] [10]

Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function

H. Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function. Seminar on Number Theory, 1979–1980 (French), Exp. No. 18, 36 pp., Univ. Bordeaux I, Talence, 1980

work page 1979

[11] [11]

Iwaniec, Topics in Classical Automorphic Forms, Grad

H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math., vol 17, Amer. Math. Soc., 1997

work page 1997

[12] [12]

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

[13] [13]

Iwaniec and Xiaoqing Li, The orthogonality of Hecke eigenvalues

H. Iwaniec and Xiaoqing Li, The orthogonality of Hecke eigenvalues. Compos. Math. 143 (2007), no. 3, 541--565

work page 2007

[14] [14]

Jutila, On spectral large sieve inequalities

M. Jutila, On spectral large sieve inequalities. Funct. Approx. Comment. Math. 28 (2000), 7-–18

work page 2000

[15] [15]

Jutila and Y

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

work page 2005

[16] [16]

Khan and M

R. Khan and M. Young, Moments and hybrid subconvexity for symmetric-square L -functions. Preprint, 2020, arXiv:2009.08419. To appear in J. Inst. Math. Jussieu

work page arXiv 2020

[17] [17]

K ral, I

E. K ral, I. Petrow, and M. Young, Oscillatory integrals with uniformity in parameters. J. Th\' e or. Nombres Bordeaux 31 (2019), no. 1, 145--159

work page 2019

[18] [18]

J. W. C. Lam, The second moment of the central values of the symmetric square L-functions. Ramanujan J. 38 (2015), no. 1, 129--145

work page 2015

[19] [19]

Lapid and W

E. Lapid and W. M\" u ller,

work page

[20] [20]

Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms

W. Luo, The spectral mean value for linear forms in twisted coefficients of cusp forms. Acta Arith. 70 (1995), no. 4, 377--391

work page 1995

[21] [21]

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

[22] [22]

Soudry, On Langlands functoriality from classical groups to GL_n

D. Soudry, On Langlands functoriality from classical groups to GL_n . Astérisque, no. 298 (2005), 335--390

work page 2005

[23] [23]

Thorner and A

J. Thorner and A. Zaman, An unconditional GL_n large sieve, Adv. Math. 378 (2021), 107529, 24 pp

work page 2021

[24] [24]

Young, The second moment of GL(3) GL(2) L -functions at special points

M. Young, The second moment of GL(3) GL(2) L -functions at special points. Math. Ann. 356 (2013), no. 3, 1005--1028

work page 2013

[25] [25]

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

[26] [26]

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

M. Young, An improved spectral large sieve inequality for SL_3( Z ) . Preprint, 2021, arXiv:2102.02796

work page internal anchor Pith review Pith/arXiv arXiv 2021