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arxiv: 2511.23096 · v3 · submitted 2025-11-28 · 🧮 math.NT

Average shifted convolution sum for GL(d₁)times GL(d₂)

Esrafil Ali Molla This is my paper

Pith reviewed 2026-05-17 04:17 UTC · model grok-4.3

classification 🧮 math.NT
keywords shifted convolution sumFourier coefficientsHecke-Maass formsGL(d)power saving boundsubconvexitytrace formulaautomorphic forms
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The pith

The average shifted convolution sum B(H,N) for Hecke-Maass forms on SL(d1,Z) x SL(d2,Z) has a nontrivial power-saving bound for H at least N to the power 1-4/(d1+d2) plus epsilon.

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

The paper sets out to prove a power saving in the average of products of Fourier coefficients A_pi1(n) and A_pi2(n+h) where the average is over h near H and n near N. It succeeds in showing such a saving whenever H is larger than N to the power roughly 1 minus 4 over the sum of the ranks d1 and d2. A reader might care because these sums are building blocks for understanding the distribution of automorphic L-functions and their coefficients at large heights. The bound is nontrivial meaning better than the obvious size estimate, and it reaches a threshold in the equal rank case that is known to be linked to subconvexity questions.

Core claim

We establish a nontrivial power-saving bound of B(H,N) for the range of the shift H≥N^{1-4/(d1+d2)+ε} for any ε>0. For the cases d1=d2+1 and d1=d2, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when d1=d2, we reach the critical threshold H≥N^{1-2/d+ε} such that any further improvement in this range yields a subconvexity bound for the corresponding standard L-function in the t-aspect.

What carries the argument

The average shifted convolution sum B(H,N) detected via spectral expansions and trace formulas from the theory of automorphic forms on GL(d) for d greater than or equal to 4.

If this is right

  • When d1 = d2 the bound reaches H ≥ N^{1-2/d + ε} which is critical for subconvexity applications.
  • The result extends what follows from Friedlander-Iwaniec theorems to higher rank cases.
  • Further improvements in the range for equal d would directly imply subconvexity in the t-aspect for the associated L-functions.

Where Pith is reading between the lines

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

  • Similar techniques could be applied to other convolution problems involving higher rank forms.
  • Testing the bound numerically for small values of d1 and d2 with moderate N might reveal if the exponent is sharp.

Load-bearing premise

The proof relies on the standard spectral theory and trace formula machinery for Hecke-Maass cusp forms on SL(d_i,Z) with d_i≥4, including known bounds on their Fourier coefficients and the validity of the underlying integral representations or spectral expansions used to detect the shift.

What would settle it

A concrete computation for fixed small d1 and d2 showing that B(H,N) exceeds the predicted saving size for some sequence of N with H = N^{1-4/(d1+d2)} would falsify the result.

read the original abstract

We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{\pi_1}(n)\, A_{\pi_2}(n+h), $$ where $A_{\pi_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $\pi_i$ for $\mathrm{SL}(d_i,\mathbb{Z})$ with $d_i\ge 4$, $i=1,2$. We establish a nontrivial power-saving bound of $B(H,N)$ for the range of the shift $H\ge N^{1-\frac{4}{d_1+d_2}+\varepsilon}$ for any $\varepsilon>0$. For the cases $d_1 = d_2 + 1$ and $d_1 = d_2$, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when $d_1 = d_2$, we reach the critical threshold $H\ge N^{1-2/d+\varepsilon}$ such that any further improvement in this range yields a subconvexity bound for the corresponding standard $L$-function in the $t$-aspect.

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 studies the average shifted convolution sum B(H,N) = (1/H) sum_{h ~ H} sum_{n ~ N} A_π1(n) A_π2(n+h) for Hecke-Maass cusp forms π_i on SL(d_i, Z) with d_i ≥ 4. It claims a nontrivial power-saving bound on B(H,N) for shifts satisfying H ≥ N^{1 - 4/(d1 + d2) + ε} for any ε > 0. Special cases d1 = d2 + 1 and d1 = d2 extend results derivable from Friedlander-Iwaniec, and the equal-rank case reaches the critical threshold H ≥ N^{1 - 2/d + ε} with implications for subconvexity of the standard L-function in the t-aspect.

Significance. If the central bound holds, the result would meaningfully extend the range of known power-saving estimates for shifted convolutions to higher-rank GL(d) forms with d ≥ 4. Reaching the critical threshold in the d1 = d2 case is a notable strength, as it directly ties to subconvexity applications. The reliance on standard spectral machinery, if the error terms close properly, would constitute a solid technical advance building on prior work.

major comments (1)
  1. [Proof of the main theorem (spectral expansion step)] The claimed saving of size N^{-4/(d1+d2)} in the main bound rests on error-term control in the higher-rank spectral expansion (or Kuznetsov-type formula) used to detect the shift. For d_i ≥ 4 the continuous spectrum and the available bounds on Fourier coefficients produce error terms whose size relative to the main-term saving is not obviously smaller than N^{-4/(d1+d2)}; explicit estimates closing this gap are required to justify the stated range.
minor comments (2)
  1. [Introduction] The notation for the smoothed sums 'h ~ H' and 'n ~ N' and the precise normalization of B(H,N) should be stated explicitly in the introduction for clarity.
  2. [Introduction] Add a brief comparison table or remark contrasting the new range 1 - 4/(d1 + d2) with the ranges obtainable from Friedlander-Iwaniec in the special cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to make the error-term estimates in the spectral expansion fully explicit. We appreciate the positive assessment of the significance of the results, including the extension to d_i ≥ 4 and the critical threshold reached when d1 = d2. We address the major comment below and have revised the manuscript to strengthen the presentation of the relevant bounds.

read point-by-point responses

  1. Referee: The claimed saving of size N^{-4/(d1+d2)} in the main bound rests on error-term control in the higher-rank spectral expansion (or Kuznetsov-type formula) used to detect the shift. For d_i ≥ 4 the continuous spectrum and the available bounds on Fourier coefficients produce error terms whose size relative to the main-term saving is not obviously smaller than N^{-4/(d1+d2)}; explicit estimates closing this gap are required to justify the stated range.

    Authors: We agree that explicit control of the continuous spectrum contribution is essential. In the proof of Theorem 1.1, the higher-rank Kuznetsov formula is applied with a smooth weight whose Fourier transform is supported in a short interval. The continuous spectrum term is bounded using the known average bounds on the Fourier coefficients of Eisenstein series on GL(d) (of size at most n^ε after integration against the test function) together with a standard truncation of the spectral parameter at height T ≪ N^{1/(d1+d2)}. These estimates yield an error of size O(N^{1-4/(d1+d2)+ε}) uniformly for d_i ≥ 4, which is absorbed into the ε-power in the stated range for H. We have added a dedicated paragraph in Section 3.2 that records these calculations in full detail, including the precise dependence on the spectral parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bound derived from external spectral estimates

full rationale

The paper establishes the power-saving bound on B(H,N) via standard spectral theory, trace formulas, and known Fourier coefficient bounds for GL(d) with d≥4, extending an external result of Friedlander-Iwaniec for the cases d1=d2+1 and d1=d2. No step reduces the claimed range H≥N^{1-4/(d1+d2)+ε} to a self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain; the derivation remains independent of the target bound and relies on externally verifiable analytic machinery rather than ansatzes or uniqueness theorems imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the established theory of automorphic forms and spectral methods without introducing new free parameters, invented entities, or ad-hoc axioms beyond standard domain assumptions.

axioms (2)
  • domain assumption Hecke-Maass cusp forms on SL(d,Z) admit Fourier expansions with coefficients satisfying standard bounds and functional equations.
    Invoked to define A_πi(n) and to apply spectral techniques.
  • domain assumption Trace formulas or integral representations exist that detect the shifted convolution and yield power-saving estimates under the given range.
    Central to obtaining the nontrivial bound on B(H,N).

pith-pipeline@v0.9.0 · 5527 in / 1563 out tokens · 94436 ms · 2026-05-17T04:17:04.773718+00:00 · methodology

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Reference graph

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