[Paper Review] Triple correlations of Fourier coefficients of cusp forms

Yongxiao Lin|arXiv (Cornell University)|Jul 11, 2016

Analytic Number Theory Research3 references3 citations

TL;DR

This paper establishes a nontrivial upper bound for triple correlations of Fourier coefficients of holomorphic cusp forms using spectral methods and the Kuznetsov trace formula. It shows that the sum $\sum_{H\leq h\leq 2H}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}\lambda_1(n-h)\lambda_2(n)\lambda_3(n+h)$ is bounded by $X^{\varepsilon}\min\left(XH,\frac{X^2}{H^{1/2}}\right)$, which is nontrivial when $H \geq X^{2/3+\varepsilon}$, providing a cuspidal analogue to Blomer's work on divisor functions.

ABSTRACT

We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms $\sum_{H\leq h\leq 2H}W\big(\frac{h}{H}\big)\sum_{X\leq n\leq 2X}λ_{1}(n-h)λ_{2}(n)λ_{3}(n+h)$, which is nontrivial provided that $H\geq X^{2/3+\varepsilon}$. The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.

Motivation & Objective

• To establish a cuspidal analogue of Blomer's result on triple correlations of divisor functions.
• To analyze unbalanced shifted convolution sums of three Hecke eigenvalues of holomorphic cusp forms.
• To derive a nontrivial upper bound for the triple correlation sum under the condition $H \geq X^{2/3+\varepsilon}$.
• To overcome technical challenges arising from the lack of a decomposition for cusp form Fourier coefficients, unlike divisor functions.

Proposed method

• Application of the Kuznetsov trace formula to relate the sum to spectral sums over Maass forms.
• Use of the circle method and Voronoi summation to handle the shifted convolution structure.
• Estimation of spectral sums via large sieve inequalities and bounds on Bessel functions.
• Control of oscillatory integrals through stationary phase and exponential sum estimates.
• Use of the $\ell^2$-norm of the coefficient sequence to control error terms in spectral expansions.
• Adaptation of Blomer’s approach for divisor functions to the cusp form setting, with modifications due to the absence of a multiplicative decomposition.

Experimental results

Research questions

• RQ1 Can a nontrivial upper bound be established for triple correlations of Fourier coefficients of holomorphic cusp forms?
• RQ2 How does the range of $H$ for nontriviality compare to the divisor function case?
• RQ3 What are the main analytic obstructions in extending Blomer’s method from Eisenstein series to cusp forms?
• RQ4 Can spectral methods and the Kuznetsov trace formula yield effective bounds in the absence of a multiplicative structure?

Key findings

• The sum $\sum_{h}W\left(\frac{h}{H}\right)\sum_{X\leq n\leq 2X}a(n)\lambda_1(n+h)\lambda_2(n-h)$ is bounded by $X^{\varepsilon}\frac{X}{H}\left((XH)^{1/2}+\frac{X}{H^{1/2}}\right)\|a\|_{2}$.
• The bound is nontrivial when $H \geq X^{2/3+\varepsilon}$, which is weaker than the $H \geq X^{1/3+\varepsilon}$ range for divisor functions.
• The result is derived via spectral methods and the Kuznetsov trace formula, with error terms controlled by large sieve inequalities and $\ell^2$-norm estimates.
• The key difficulty lies in the inability to decompose cusp form Fourier coefficients as in the divisor function case, limiting the range of $H$.
• The bound matches the size of the spectral terms in Blomer’s result, indicating that the spectral contribution is the dominant error source.