Triple correlations of Fourier coefficients of cusp forms

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 ${\sum }_{H\le h\le 2H}W\left(\frac{h}{H}\right){\sum }_{X\le n\le 2X}{\lambda }_1(n-h){\lambda }_2(n){\lambda }_3(n+h)$, which is nontrivial provided that $H\ge X^{2/3+\epsilon }$. The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.

12 pages

Cited by in corpus (2)

• On double shifted convolution sum of $SL(2,\mathbb{Z})$ Hecke eigen forms
• Averages of shifted convolution sums for $GL(3)\times GL(2)$