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Shifted convolution sum with weighted average : $GL(3) \times GL(3)$ setup

Mohd Harun, Saurabh Kumar Singh

TL;DR

This work develops non-trivial bounds for both the average and weighted-average shifted convolution sums in a GL(3) $\times$ GL(3) setting, using a Duke–Friedlander–Iwaniec delta method to separate oscillations and a sequence of GL(2) and GL(3) Voronoi summations. The analysis hinges on delicate control of zero- and non-zero-frequency contributions, aided by Poisson summation and square-root cancellations in intricate character sums built from Kloosterman-type objects, with key technical input from the geometry of Kloosterman sheaves. The main results provide bounds $\mathcal{D}(H,X) \ll X^{1-\delta+\epsilon}$ and $\mathcal{L}(H,X) \ll X^{1-\delta+\epsilon}$ for $X^{1/2+\delta} \le H \le X$, significantly advancing understanding of higher-rank shifted convolution problems and their connections to subconvexity and higher-degree $L$-function moments. The techniques combine advanced summation formulas, harmonic analysis, and deep algebraic-geometric input on exponential sums, with potential impact on subconvexity problems for GL(3) and beyond.

Abstract

This article will prove non-trivial estimates for the average and weighted average version of general $GL(3) \times GL(3)$ shifted convolution sums by using the circle method.

Shifted convolution sum with weighted average : $GL(3) \times GL(3)$ setup

TL;DR

This work develops non-trivial bounds for both the average and weighted-average shifted convolution sums in a GL(3) GL(3) setting, using a Duke–Friedlander–Iwaniec delta method to separate oscillations and a sequence of GL(2) and GL(3) Voronoi summations. The analysis hinges on delicate control of zero- and non-zero-frequency contributions, aided by Poisson summation and square-root cancellations in intricate character sums built from Kloosterman-type objects, with key technical input from the geometry of Kloosterman sheaves. The main results provide bounds and for , significantly advancing understanding of higher-rank shifted convolution problems and their connections to subconvexity and higher-degree -function moments. The techniques combine advanced summation formulas, harmonic analysis, and deep algebraic-geometric input on exponential sums, with potential impact on subconvexity problems for GL(3) and beyond.

Abstract

This article will prove non-trivial estimates for the average and weighted average version of general shifted convolution sums by using the circle method.
Paper Structure (16 sections, 23 theorems, 207 equations)

This paper contains 16 sections, 23 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. DFI Delta Method
  3. Sketch of the proof of Theorem 1
  4. Preliminaries
  5. Proof of Theorem \ref{['th1']}
  6. Applying the Voronoi summation formulae
  7. Simplification of integrals
  8. Applying Cauchy-Schwarz inequality
  9. Applying Poisson summation formula
  10. Analysis of integral $\mathcal{J}(h,n_2)$
  11. Analysis of character sum $\mathfrak{C}(q;h,n_2)$
  12. Estimation of zero frequency
  13. Estimation of non-zero frequency
  14. Final Estimates
  15. Proof of Lemma \ref{['FK']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathcal{D}(H, X)$ be as defined above. For any $\epsilon > 0$ and $X^{1/2 + \delta} \leq H < X^{1-\epsilon }$ with $\delta > 0$, we have where the implied constants are depending upon $\pi_1, \pi_2$,$f$ and $\epsilon$.

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • ...and 34 more