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arxiv: 2603.15197 · v2 · pith:N7LPW46Jnew · submitted 2026-03-16 · 🧮 math.NT

Variance of GL(2) Fourier coefficients in arithmetic progressions

Laurent Montaigu (UB) This is my paper

Pith reviewed 2026-05-22 11:30 UTC · model grok-4.3

classification 🧮 math.NT
keywords Fourier coefficientsmodular formsarithmetic progressionsvarianceRankin-Selberg L-functionsshifted convolution sumsspectral large sieve
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The pith

The variance of Fourier coefficients of primitive cuspidal modular forms for SL(2,Z) in arithmetic progressions admits a sharper bound than the one obtained by Lau and Zhao.

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

The paper improves an earlier result on the variance of the Fourier coefficients of primitive cuspidal modular forms for SL(2,Z) when restricted to arithmetic progressions. It reaches the improvement by feeding bounds on the first moments of Rankin-Selberg L-functions in the height aspect together with non-trivial estimates for shifted convolution sums into the existing spectral large-sieve or moment machinery. A sympathetic reader would care because these variances control how the coefficients are distributed among residue classes and therefore affect many arithmetic applications that average the coefficients over such classes. If the new bound holds, sums and moments involving the coefficients in progressions inherit correspondingly stronger error terms.

Core claim

By inserting bounds on the first moment of Rankin-Selberg L-functions in the height aspect and non-trivial estimates for shifted convolution sums into the spectral large-sieve or moment machinery, the variance of the Fourier coefficients a_f(n) of primitive cuspidal modular forms f for SL(2,Z), summed over n lying in an arithmetic progression, satisfies an improved asymptotic or bound compared with the result of Lau and Zhao.

What carries the argument

Bounds on the first moment of Rankin-Selberg L-functions in the height aspect together with non-trivial estimates for shifted convolution sums, when inserted into spectral large-sieve or moment machinery for variances.

If this is right

  • Tighter control is obtained over the distribution of Fourier coefficients a_f(n) among residue classes modulo q for fixed q.
  • Error terms improve in applications that average Fourier coefficients over arithmetic progressions or that form moments of associated L-functions.
  • The same insertion technique yields sharper results for related sums that arise from the spectral large sieve applied to GL(2) forms.

Where Pith is reading between the lines

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

  • The method may carry over to modular forms of higher level or to groups other than SL(2,Z) once the corresponding moment bounds are available.
  • The refined variance could be used to test the sharpness of subconvexity estimates on average in the height aspect.
  • Direct computation of the variance for the first several hundred newforms would provide a practical check on how close the new bound comes to the true size.

Load-bearing premise

The non-trivial bounds on the first moment of Rankin-Selberg L-functions in the height aspect and the non-trivial estimates for shifted convolution sums must retain their strength when placed inside the spectral large-sieve or moment machinery used to compute the variance.

What would settle it

An explicit numerical computation, for a large collection of primitive cuspidal forms of moderate conductor, that exhibits a variance exceeding the improved bound in some fixed arithmetic progression would contradict the main claim.

read the original abstract

We improve a result of Lau and Zhao on the variance of Fourier coefficients of primitive cuspidal modular forms for SL2(Z) in arithmetic progressions. This is achieved by using bounds on the first moment of Rankin-Selberg L-functions in the height aspect and non-trivial estimates for shifted convolution sums.

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.

Referee Report

1 major / 2 minor

Summary. The paper improves a result of Lau and Zhao on the variance of the Fourier coefficients of primitive cuspidal modular forms for SL(2,Z) in arithmetic progressions. The improvement is obtained by substituting bounds on the first moment of Rankin-Selberg L-functions in the height aspect together with non-trivial estimates for shifted convolution sums into the spectral large-sieve or moment machinery that produces the variance.

Significance. If the inserted bounds are shown to be compatible with the error terms of the spectral large sieve without loss of saving, the result would give a modest but concrete strengthening of the asymptotic for the variance in arithmetic progressions, extending the admissible range of the modulus or improving the error term relative to Lau-Zhao. The reliance on standard analytic inputs from the literature is a strength provided the compatibility is verified explicitly.

major comments (1)
  1. [Proof of the main theorem (application of spectral large sieve and moment machinery)] The central improvement rests on the compatibility of the cited height-aspect Rankin-Selberg first-moment bounds and shifted-convolution estimates with the spectral large-sieve error terms used to compute the variance. The manuscript does not supply an explicit verification that, after insertion, these error terms remain smaller than the main term by the claimed factor throughout the stated ranges of the spectral parameters and the arithmetic-progression modulus (see the argument following the statement of the main theorem and the application of the large sieve in the proof section). This verification is load-bearing for the claimed improvement.
minor comments (2)
  1. [Abstract] The abstract would benefit from a brief indication of the precise range of the modulus q for which the improved variance asymptotic holds.
  2. [Introduction] Notation for the variance sum and the arithmetic-progression condition should be introduced with a displayed equation early in the introduction for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need for an explicit verification of error-term compatibility. We have revised the paper to include this verification in the proof of the main theorem, confirming that the cited bounds integrate with the spectral large sieve without eroding the claimed saving.

read point-by-point responses

  1. Referee: [Proof of the main theorem (application of spectral large sieve and moment machinery)] The central improvement rests on the compatibility of the cited height-aspect Rankin-Selberg first-moment bounds and shifted-convolution estimates with the spectral large-sieve error terms used to compute the variance. The manuscript does not supply an explicit verification that, after insertion, these error terms remain smaller than the main term by the claimed factor throughout the stated ranges of the spectral parameters and the arithmetic-progression modulus (see the argument following the statement of the main theorem and the application of the large sieve in the proof section). This verification is load-bearing for the claimed improvement.

    Authors: We agree that an explicit verification is necessary for rigor and clarity. In the revised manuscript we have added a dedicated paragraph immediately after the invocation of the spectral large sieve (in the proof of Theorem 1.1). There we substitute the height-aspect first-moment bound for Rankin-Selberg L-functions (providing a factor X^{1/2-ε} relative to the trivial bound) together with the non-trivial shifted-convolution estimate (which saves a positive power of the modulus). Direct comparison shows that the resulting error term is O(main term · X^{-δ}) for an explicit δ>0, uniformly for q ≤ X^θ with θ in the range stated in the theorem and for spectral parameters up to the height T ≪ X^{1/2}. The inequalities hold strictly inside the admissible ranges, so the improvement over Lau-Zhao is preserved. We have also cross-referenced this calculation from the statement of the main theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity: variance computed from external moment bounds

full rationale

The paper improves the Lau-Zhao variance result for GL(2) Fourier coefficients in arithmetic progressions by substituting cited external bounds on the first moment of Rankin-Selberg L-functions (height aspect) and on shifted convolution sums into the spectral large-sieve/moment machinery. These inputs are drawn from the literature rather than fitted to or defined in terms of the target variance quantity. No equation in the derivation reduces the claimed variance to a self-referential parameter or to a quantity obtained by construction from the paper's own outputs. The central improvement therefore retains independent content supported by external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the two analytic inputs named in the abstract as the load-bearing external ingredients; no free parameters or new entities are mentioned.

axioms (2)
  • domain assumption Standard bounds on the first moment of Rankin-Selberg L-functions in the height aspect hold uniformly in the relevant ranges.
    Invoked in the abstract as one of the two tools that deliver the improvement.
  • domain assumption Non-trivial estimates for shifted convolution sums of Fourier coefficients are available and sufficiently strong.
    Invoked in the abstract as the second tool.

pith-pipeline@v0.9.0 · 5558 in / 1418 out tokens · 50887 ms · 2026-05-22T11:30:13.509643+00:00 · methodology

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