[Paper Review] Shifted convolution sums of $GL_3$ cusp forms with $ heta$-series

Qingfeng Sun|arXiv (Cornell University)|Sep 25, 2015

Analytic Number Theory Research21 references3 citations

TL;DR

This paper establishes an unconditional bound for shifted convolution sums involving the Fourier coefficients of a $GL_3$ Hecke-Maass cusp form and the ternary theta series $r_3(n)$, using advanced exponential sum estimates and character sum analysis. The key result is a hybrid bound of $X^{3/2 - 1/8 + \varepsilon}$, improving upon conditional bounds and demonstrating nontrivial cancellation despite the absence of strong Kloosterman sum bounds.

ABSTRACT

Let $A_f(1,n)$ be the normalized Fourier coefficients of a Hecke-Maass cusp form $f$ for $SL_3(\\mathbb{Z})$ and $$ r_3(n)=\\#\\left\\{(n_1,n_2,n_3)\\in \\mathbb{Z}^3:n_1^2+n_2^2+n_3^2=n\ ight\\}. $$ Let $1\\leq h\\leq X$ and $\\phi(x)$ be a smooth function compactly supported on $[1/2,1]$. It is shown that $$ \\sum_{n\\geq 1}A_f(1,n+h)r_3(n)\\phi\\left(\\frac{n}{X}\ ight) \\ll_{f,\\varepsilon} X^{\\frac{3}{2}-\\frac{1}{8}+\\varepsilon} $$ uniformly with respect to the shift $h$.

Motivation & Objective

• To establish a nontrivial, unconditional upper bound for shifted convolution sums involving $GL_3$ Hecke-Maass cusp forms and the ternary theta series $r_3(n)$, which counts representations of integers as sums of three squares.
• To overcome the lack of strong bounds for exponential sums twisted by $r_3(n)$, which obstructs the use of Jutila’s circle method in this context.
• To generalize previous results on shifted convolution sums with $GL_2$ forms to the more complex $GL_3$ setting, particularly for $r_3(n)$, which is the Fourier coefficient of $\theta^3(z)$.
• To provide a quantitative improvement over conditional bounds derived from the Ramanujan conjecture and known estimates for $GL_3$ $L$-functions.
• To extend the method to higher $\ell$-fold theta series $r_\ell(n)$, showing the approach generalizes beyond $\ell=3$.

Proposed method

• The proof relies on Voronoi summation for $GL_3$ cusp forms and a new Voronoi-type formula for $r_3(n)$, derived from the theory of $\theta^3(z)$.
• The shifted convolution sum is transformed via Poisson summation and character sum decomposition, reducing the problem to estimating a complex character sum $\mathscr{C}(b_1,b_2,b_3,n_1,n_2,h,v;q)$ over moduli $q$.
• The character sum is further analyzed by splitting into dyadic intervals and applying Weil’s bound for Kloosterman sums and Salié sum estimates in the case of $p \mid n_2$.
• For the generic case, the method uses the nondegeneracy of the associated Laurent polynomial $f(x,y)$ with respect to its Newton polyhedron $\Delta(f)$, applying results from exponential sum theory over finite fields.
• The key technical step involves bounding a twisted Ramanujan sum via the $\epsilon$-removal technique, leveraging the nondegeneracy of $f$ and the interior position of the origin in $\Delta(f)$.
• The final bound is obtained by combining $L^2$-type estimates and $\ell^2$-concentration arguments, leading to a power-saving gain of $X^{-1/8 + \varepsilon}$ over the trivial bound.

Experimental results

Research questions

• RQ1 Can a nontrivial, unconditional upper bound be established for shifted convolution sums of the form $\sum_n A_f(1,n+h) r_3(n) \phi(n/X)$, where $A_f$ is a $GL_3$ Hecke-Maass cusp form and $r_3(n)$ counts three-square representations?
• RQ2 Why do standard methods like Jutila’s circle method fail in this $GL_3$-to-$\theta$-series setting, and can alternative techniques overcome this obstruction?
• RQ3 What is the optimal exponent in the hybrid bound $X^{3/2 - \delta + \varepsilon}$ for such sums, and can it be improved beyond the conditional estimates based on the Ramanujan conjecture?
• RQ4 How does the structure of the Newton polyhedron $\Delta(f)$ of the associated exponential sum influence the cancellation in the character sum?
• RQ5 Can the method be extended to $r_\ell(n)$ for $\ell \geq 4$, and does the exponent $\delta = 1/8$ persist in higher dimensions?

Key findings

• The paper establishes the unconditional bound $\sum_{n \geq 1} A_f(1,n+h) r_3(n) \phi(n/X) \ll_{f,\varepsilon} X^{3/2 - 1/8 + \varepsilon}$ for $1 \leq h \leq X$, which is stronger than the conditional bound $X^{3/2 + \varepsilon}$ under the Ramanujan conjecture.
• The bound is achieved through a novel analysis of a triple character sum $\mathscr{C}(b_1,b_2,b_3,n_1,n_2,h,v;q)$, where the key step involves proving $\widetilde{\mathscr{T}}(p) \ll (h,p)^{1/2} p$ via nondegeneracy of the associated Laurent polynomial over $\mathbb{F}_p^\times$.
• The method avoids reliance on Kloosterman sum bounds for $r_3(n)$, which are insufficient for Jutila’s circle method, and instead uses $\epsilon$-removal via Newton polyhedra and exponential sum estimates.
• The bound is uniform in the shift $h$, and the method generalizes to $r_\ell(n)$ for $\ell \geq 4$, with the exponent $-1/8$ expected to persist in higher dimensions.
• For the special case where $A_f(1,n)$ is replaced by the triple divisor function $\tau_3(n)$, the exponent can be improved beyond $-1/8$ due to the $n^\varepsilon$-bound on $\tau_3(n)$, yielding a better saving.
• The result demonstrates that even in the absence of strong $GL_2$-type exponential sum estimates, deep cancellation can still be extracted from $GL_3$-to-$\theta$-series convolution sums via algebraic geometry and character sum techniques.