On shifted convolution sums of $\mathrm{GL}(3)$-Fourier coefficients with an average over shifts

TL;DR

The paper examines the average shifted convolution of GL$(3)$ Fourier coefficients attached to a Hecke–Maass form, formulating the problem as a two-variable sum $S'$. It combines the delta-method with conductor lowering, Voronoi summation on GL$(3)$-coefficients, Poisson summation, Mellin–Barnes linearization, and an iterative Cauchy-doubling strategy to extract power-saving cancellations across a broad range of the shift parameter $H$. The main result provides a nontrivial bound for $S'$ in the regime $N^{1/2-\varepsilon}\ge H\ge N^{1/6+\varepsilon}$, improving prior work that required $H\ge N^{1/4+\varepsilon}$. The approach hinges on carefully disentangling oscillations, controlling character sums, and repeatedly dualizing sums to reduce conductors, ultimately yielding a sharpened upper bound that advances subconvex-type cancellation questions for ${\mathrm{GL}}(3)\times{\mathrm{GL}}(3)$-type shifted sums with an averaging over shifts.

Abstract

Let $F$ be a Hecke-Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ and $A(m,n)$ be its normalized Fourier coefficients. Let $V$ be a smooth function, compactly supported on $[1,2]$ and satisfying $V(y)^{j} \ll_j y^{-j}$ for any $j \in \mathbb{N} \cup \{0\}$. In this article we prove a power-saving upper bound for the `average' shifted convolution sum \begin{equation*} \sum_{h}\sum_{n}A(1,n)A(1,n+h)V\left(\frac{n}{N}\right)V\left(\frac{h}{H}\right), \end{equation*} for the range $N^{1/2-\varepsilon} \geq H \geq N^{1/6+ \varepsilon}$, for any $\varepsilon >0$. This is an improvement over the previously known range $N^{1/2-\varepsilon} \geq H \geq N^{1/4+ \varepsilon}$.

Theorems & Definitions (43)

• Theorem 1
• Remark 1
• Remark 2
• Lemma 2.1: Ramanujan bound on average, molteni_upper_2002
• Lemma 2.2
• proof
• Lemma 2.3: Voronoi type summation formula
• Lemma 2.4
• Lemma 2.4
• Lemma 2.5