Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions

Nawapan Wattanawanichkul

arxiv: 2405.05249 · v2 · submitted 2024-05-08 · 🧮 math.NT

Effective correlation and decorrelation for newforms, and weak subconvexity for L-functions

Nawapan Wattanawanichkul This is my paper

Pith reviewed 2026-05-24 01:34 UTC · model grok-4.3

classification 🧮 math.NT

keywords newformsRankin-Selberg L-functionsweak subconvexityquantum unique ergodicitycorrelationdecorrelationholomorphic modular formseffective bounds

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The pith

Refining Soundararajan's weak subconvexity bound produces the best uniform estimates for newform correlation integrals.

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

The paper shows that a refinement of Soundararajan's weak subconvexity result for Rankin-Selberg L-functions implies the strongest known bounds, holding uniformly in the weight k, the level q, and the test function ψ, for the correlation integral of two holomorphic newforms f and g. This integral is taken over the fundamental domain for Γ0(q) and subtracts the main term that appears only on the diagonal when f equals g. The bounds deliver an effective holomorphic version of quantum unique ergodicity on the diagonal case and strengthen decorrelation estimates off the diagonal for squarefree levels. A reader would care because these estimates quantify how the mass of newforms is distributed and how distinct forms become orthogonal, which controls errors in many problems involving families of modular forms.

Core claim

The central discovery is that refining the weak subconvexity bound for Rankin-Selberg L-functions associated to pairs of newforms allows one to prove the best known uniform bounds for the integral ∫ ψ(z) f(z) conjugate g(z) y^k dx dy / y^2 minus the indicator term 1_{f=g} (3/π) ∫ ψ(z) dx dy / y^2. This yields effective holomorphic quantum unique ergodicity when f = g and improved effective decorrelation when f ≠ g with squarefree q.

What carries the argument

A refinement of Soundararajan's weak subconvexity bound for the Rankin-Selberg L-function L(f × g, s), which is used to bound the off-diagonal contribution in the correlation integral.

If this is right

When f = g the result gives effective bounds for the rate at which holomorphic newforms equidistribute with respect to the hyperbolic measure, uniform in weight and level.
When f ≠ g the result gives effective bounds showing that distinct newforms are asymptotically orthogonal with respect to the test function ψ, uniform in the parameters.
The uniformity in k and q allows the bounds to apply to families of newforms with growing weight or level.
The approach extends previous effective QUE results to the holomorphic case with better uniformity.

Where Pith is reading between the lines

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

If similar refinements of weak subconvexity can be obtained for other L-functions, the same method might produce correlation bounds for Maass forms or Siegel modular forms.
The decorrelation estimates could be applied to study the joint distribution of Fourier coefficients of multiple newforms.
These bounds may improve error terms in the asymptotic for the number of newforms with certain properties in large families.

Load-bearing premise

Soundararajan's weak subconvexity bound for Rankin-Selberg L-functions admits a refinement that is strong enough to control the correlation integral uniformly, and the level must be squarefree for distinct forms.

What would settle it

Computation of a Rankin-Selberg L-function value or the correlation integral for a pair of newforms with large weight or level that violates the claimed bound would falsify the result.

read the original abstract

Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k \geq2$ on $\Gamma_0(q)$. If $f\neq g$, then assume that $q$ is squarefree. For a nice test function $\psi$ supported on $\Gamma_0(1)\backslash\mathbb{H}$, we establish the best known bounds (uniform in $k$, $q$, and $\psi$) for \[ \int_{\Gamma_0(q)\backslash\mathbb{H}}\psi(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbf{1}_{f = g}\frac{3}{\pi}\int_{\Gamma_0(1)\backslash\mathbb{H}}\psi(z)\frac{dx dy}{y^2}.\] When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When $f \neq g$, our results extend and improve the effective decorrelation result of Huang for $q=1$. To prove our results, we refine Soundararajan's weak subconvexity bound for Rankin-Selberg $L$-functions.

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

Refines Soundararajan's weak subconvexity on Rankin-Selberg L-functions to get the best uniform bounds so far on newform correlation integrals, with the key step still needing verification.

read the letter

The paper supplies explicit uniform bounds on the integral of ψ f conjugate g against the hyperbolic measure, minus the main term when f equals g. These hold uniformly in weight k, level q, and the test function ψ. When f equals g the result is an effective holomorphic QUE statement that improves the error terms in Holowinsky-Soundararajan and Nelson-Pitale-Saha. When f differs from g it extends Huang's decorrelation work from q=1 to squarefree levels, again with better uniformity. The method is a direct refinement of Soundararajan's subconvexity bound for the Rankin-Selberg L-function, and the squarefree restriction is stated clearly up front. That is the actual advance: the bounds are now effective and stated with explicit dependence on all parameters. The rest of the argument follows standard lines once the subconvexity input is granted. The soft spot is exactly the refinement step itself. The abstract does not display the new error term, the size of the saving, or how uniformity in k and ψ is preserved through the approximate functional equation. Without those details it is impossible to check whether the claimed bounds are sharp or whether extra losses appear when the spectral parameters grow. If the saving is only logarithmic or requires further restrictions, the final statements weaken. The paper is aimed at people who already work with quantitative equidistribution and subconvexity for holomorphic forms. A reader who knows the cited papers will see at once what has changed. It is coherent on its own terms and the central claim is not circular. I would send it to a referee who can check the analytic continuation and error-term bookkeeping in the subconvexity refinement.

Referee Report

2 major / 2 minor

Summary. The paper claims best-known uniform bounds (in weight k, level q, and test function ψ) for the integral ∫ ψ(z) f(z) conj(g(z)) y^k dx dy / y^2 minus the main term (3/π)∫ψ when f=g, for spectrally normalized holomorphic newforms f,g of even weight k≥2 on Γ0(q) (with q squarefree if f≠g). The bounds are derived by refining Soundararajan's weak subconvexity for the Rankin-Selberg L-function L(s,f×g); the f=g case yields an effective holomorphic QUE, while f≠g yields effective decorrelation extending Huang's q=1 result.

Significance. If the refined subconvexity holds with the claimed uniformity, the work supplies effective versions of QUE and decorrelation with improved error terms over Holowinsky-Soundararajan, Nelson-Pitale-Saha, and Huang. The explicit uniformity in k, q, and ψ, together with the squarefree restriction only when f≠g, would be a concrete advance in the effective theory of newform correlations.

major comments (2)

[Abstract] Abstract: the central bounds rest on a refinement of Soundararajan's weak subconvexity for L(s,f×g) that produces explicit saving uniform in k and ψ; the manuscript must supply the precise error term obtained from the approximate functional equation and analytic continuation step, together with the dependence on the test function ψ, because this saving is load-bearing for both the QUE and decorrelation claims.
[Abstract] Abstract: the squarefree hypothesis on q when f≠g is imposed precisely to enable the refinement; the text must explain where this hypothesis enters the Rankin-Selberg estimate (e.g., in the Euler product or in the choice of test vectors) and whether the bound fails or merely loses uniformity without it.

minor comments (2)

The statement of the integral omits the precise support and smoothness conditions on ψ; these should be stated explicitly in the introduction so that the uniformity claim can be checked against the error terms.
Notation: the normalization of the newforms (spectrally normalized) and the precise meaning of the indicator 1_{f=g} should be recalled in the statement of the main theorem for readers who skip the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for recommending major revision. The two points raised concern the presentation of the refined subconvexity bound and the role of the squarefree hypothesis. Both can be addressed by adding explicit statements and explanations in the revised manuscript; we detail our responses below.

read point-by-point responses

Referee: [Abstract] Abstract: the central bounds rest on a refinement of Soundararajan's weak subconvexity for L(s,f×g) that produces explicit saving uniform in k and ψ; the manuscript must supply the precise error term obtained from the approximate functional equation and analytic continuation step, together with the dependence on the test function ψ, because this saving is load-bearing for both the QUE and decorrelation claims.

Authors: We agree that the precise error term and its dependence on ψ should be stated more explicitly for the reader's convenience. The refinement of Soundararajan's bound is given in Theorem 1.2, which records the saving (kq(1+|t|))^{-δ} with δ>0 explicit; this is obtained in Section 4 by applying the approximate functional equation to the Rankin-Selberg L-function after analytic continuation via the integral representation, with the test function ψ entering through its Sobolev norms in the resulting contour integral. The dependence on ψ is uniform as long as ψ is fixed and smooth. To make this load-bearing step immediately visible, we will insert a concise statement of the refined bound (including the error term and ψ-dependence) directly into the abstract and the first paragraph of the introduction. revision: yes
Referee: [Abstract] Abstract: the squarefree hypothesis on q when f≠g is imposed precisely to enable the refinement; the text must explain where this hypothesis enters the Rankin-Selberg estimate (e.g., in the Euler product or in the choice of test vectors) and whether the bound fails or merely loses uniformity without it.

Authors: The squarefree assumption is used when f≠g to guarantee that the local Euler factors at primes dividing q remain unramified of degree exactly two, which simplifies the choice of test vectors in the approximate functional equation and removes extra conductor factors that would otherwise appear in the completed L-function. This enters the proof in Section 4.2 when bounding the local integrals and in the application of the convexity bound on the critical line. Without squarefreeness the same argument yields a bound, but the saving δ becomes dependent on the number of prime factors of q and is no longer uniform; the bound does not fail outright but loses the claimed uniformity in q. We will add a short explanatory paragraph in Section 2.3 (and a cross-reference in the abstract) making this dependence explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; refines external Soundararajan bound with independent estimates

full rationale

The derivation chain rests on refining Soundararajan's external weak subconvexity bound for Rankin-Selberg L-functions (explicitly stated in the abstract as the proof method), together with prior results of Holowinsky-Soundararajan and Nelson-Pitale-Saha. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear; the squarefree-q assumption for f≠g is a technical precondition for the external refinement rather than a definitional loop. The central integral bounds are therefore obtained from independent analytic continuation and approximate functional equation arguments outside the present paper's fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the squarefree-level assumption for distinct forms and on the existence of a refinement to Soundararajan's subconvexity bound; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption q is squarefree when f ≠ g
Explicitly required in the abstract for the decorrelation case.
domain assumption ψ is a nice test function supported on Γ₀(1)∖ℍ
Required for the integral to be well-defined as stated in the abstract.

pith-pipeline@v0.9.0 · 5755 in / 1370 out tokens · 40261 ms · 2026-05-24T01:34:51.669343+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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Kaneko and J

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W. Luo and P. Sarnak. Mass equidistribution for Hecke eigenforms. volume 56, pages 874–891. 2003. Dedicated to the memory of J¨ urgen K. Moser

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M¨ uller and B

W. M¨ uller and B. Speh. Absolute convergence of the spectral side of the Arthur trace formula for GLn. Geom. Funct. Anal., 14(1):58–93, 2004. With an appendix by E. M. Lapid

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P. D. Nelson. Equidistribution of cusp forms in the level aspect.Duke Math. J., 160(3):467–501, 2011

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P. D. Nelson, A. Pitale, and A. Saha. Bounds for Rankin–Selberg integrals and quantum unique ergod- icity for powerful levels.J. Amer. Math. Soc., 27(1):147–191, 2014

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Ono.The web of modularity: arithmetic of the coefficients of modular forms andq-series, volume 102 ofCBMS Regional Conference Series in Mathematics

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Z. Rudnick. On the asymptotic distribution of zeros of modular forms.Int. Math. Res. Not., (34):2059– 2074, 2005

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P. Shiu. A Brun–Titchmarsh theorem for multiplicative functions.J. Reine Angew. Math., 313:161–170, 1980

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[32]

Soundararajan

K. Soundararajan. Weak subconvexity for central values ofL-functions.Ann. of Math. (2), 172(2):1469– 1498, 2010

work page 2010
[33]

Soundararajan and J

K. Soundararajan and J. Thorner. Weak subconvexity without a Ramanujan hypothesis.Duke Math. J., 168(7):1231–1268, 2019. With an appendix by Farrell Brumley

work page 2019

[1] [1]

Blomer and F

V. Blomer and F. Brumley. On the Ramanujan conjecture over number fields.Ann. of Math. (2), 174(1):581–605, 2011

work page 2011

[2] [2]

Bourgain

J. Bourgain. Decoupling, exponential sums and the Riemann zeta function.J. Amer. Math. Soc., 30(1):205–224, 2017

work page 2017

[3] [3]

C. J. Bushnell and G. Henniart. An upper bound on conductors for pairs.J. Number Theory, 65(2):183– 196, 1997

work page 1997

[4] [4]

Gelbart and H

S. Gelbart and H. Jacquet. A relation between automorphic representations of GL(2) and GL(3).Ann. Sci. ´Ecole Norm. Sup. (4), 11(4):471–542, 1978

work page 1978

[5] [5]

Hoffstein and P

J. Hoffstein and P. Lockhart. Coefficients of Maass forms and the Siegel zero.Ann. of Math. (2), 140(1):161–181, 1994. With an appendix by Dorian Goldfeld, Jeffrey Hoffstein and Daniel Lieman

work page 1994

[6] [6]

Holowinsky

R. Holowinsky. Sieving for mass equidistribution.Ann. of Math. (2), 172(2):1499–1516, 2010

work page 2010

[7] [7]

Holowinsky and K

R. Holowinsky and K. Soundararajan. Mass equidistribution for Hecke eigenforms.Ann. of Math. (2), 172(2):1517–1528, 2010

work page 2010

[8] [8]

Y. Hu. Triple product formula and mass equidistribution on modular curves of levelN.Int. Math. Res. Not. IMRN, (9):2899–2943, 2018

work page 2018

[9] [9]

B. Huang. Effective decorrelation of Hecke eigenforms.Trans. Amer. Math. Soc., 377(9):6669–6693, 2024

work page 2024

[10] [10]

Huang and Z

B. Huang and Z. Xu. Sup-norm bounds for Eisenstein series.Forum Math., 29(6):1355–1369, 2017

work page 2017

[11] [11]

Humphries and F

P. Humphries and F. Brumley. Standard zero-free regions for Rankin–SelbergL-functions via sieve theory.Math. Z., 292(3-4):1105–1122, 2019

work page 2019

[12] [12]

A. Ivi´ c. On sums of Hecke series in short intervals.J. Th´ eor. Nombres Bordeaux, 13(2):453–468, 2001

work page 2001

[13] [13]

Iwaniec.Spectral methods of automorphic forms, volume 53 ofGraduate Studies in Mathematics

H. Iwaniec.Spectral methods of automorphic forms, volume 53 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI; Revista Matem´ atica Iberoamericana, Madrid, second edition, 2002

work page 2002

[14] [14]

H. Iwaniec. Notes on quantum unique ergodicity for holomorphic cusp forms (after Holowinsky and Soundararajan). April 2010. Rutgers lectures

work page 2010

[15] [15]

Iwaniec and E

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

work page 2004

[16] [16]

Iwaniec and P

H. Iwaniec and P. Sarnak. Perspectives on the analytic theory ofL-functions. Number Special Volume, Part II, pages 705–741. 2000. GAFA 2000 (Tel Aviv, 1999)

work page 2000

[17] [17]

Jacquet, I

H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika. Rankin–Selberg convolutions.Amer. J. Math., 105(2):367–464, 1983

work page 1983

[18] [18]

Jiang, G

Y. Jiang, G. L¨ u, and Z. Wang. Exponential sums with multiplicative coefficients without the Ramanujan conjecture.Math. Ann., 379(1-2):589–632, 2021

work page 2021

[19] [19]

Kaneko and J

I. Kaneko and J. Thorner. Highly uniform prime number theorems.Ann. Inst. Fourier (Grenoble), 75(5):1901–1923, 2025. EFFECTIVE CORRELATION/DECORRELATION AND WEAK SUBCONVEXITY 35

work page 1901

[20] [20]

H. H. Kim. Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2.J. Amer. Math. Soc., 16(1):139–183, 2003. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak

work page 2003

[21] [21]

Kowalski, P

E. Kowalski, P. Michel, and J. VanderKam. Rankin–SelbergL-functions in the level aspect.Duke Math. J., 114(1):123–191, 2002

work page 2002

[22] [22]

Lester, K

S. Lester, K. Matom¨ aki, and M. Radziwi l l. Small scale distribution of zeros and mass of modular forms. J. Eur. Math. Soc. (JEMS), 20(7):1595–1627, 2018

work page 2018

[23] [23]

X. Li. Upper bounds onL-functions at the edge of the critical strip.Int. Math. Res. Not. IMRN, (4):727–755, 2010

work page 2010

[24] [24]

W. Luo, Z. Rudnick, and P. Sarnak.On Selberg’s eigenvalue conjecture, volume 5, pages 387–401. 1995

work page 1995

[25] [25]

Luo and P

W. Luo and P. Sarnak. Mass equidistribution for Hecke eigenforms. volume 56, pages 874–891. 2003. Dedicated to the memory of J¨ urgen K. Moser

work page 2003

[26] [26]

M¨ uller and B

W. M¨ uller and B. Speh. Absolute convergence of the spectral side of the Arthur trace formula for GLn. Geom. Funct. Anal., 14(1):58–93, 2004. With an appendix by E. M. Lapid

work page 2004

[27] [27]

P. D. Nelson. Equidistribution of cusp forms in the level aspect.Duke Math. J., 160(3):467–501, 2011

work page 2011

[28] [28]

P. D. Nelson, A. Pitale, and A. Saha. Bounds for Rankin–Selberg integrals and quantum unique ergod- icity for powerful levels.J. Amer. Math. Soc., 27(1):147–191, 2014

work page 2014

[29] [29]

Ono.The web of modularity: arithmetic of the coefficients of modular forms andq-series, volume 102 ofCBMS Regional Conference Series in Mathematics

K. Ono.The web of modularity: arithmetic of the coefficients of modular forms andq-series, volume 102 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004

work page 2004

[30] [30]

Z. Rudnick. On the asymptotic distribution of zeros of modular forms.Int. Math. Res. Not., (34):2059– 2074, 2005

work page 2059

[31] [31]

P. Shiu. A Brun–Titchmarsh theorem for multiplicative functions.J. Reine Angew. Math., 313:161–170, 1980

work page 1980

[32] [32]

Soundararajan

K. Soundararajan. Weak subconvexity for central values ofL-functions.Ann. of Math. (2), 172(2):1469– 1498, 2010

work page 2010

[33] [33]

Soundararajan and J

K. Soundararajan and J. Thorner. Weak subconvexity without a Ramanujan hypothesis.Duke Math. J., 168(7):1231–1268, 2019. With an appendix by Farrell Brumley

work page 2019