Triple Correlation Sums of Coefficients of Maass Forms For SL4(ℤ)

Abstract

ABSTRACT We investigate the triple correlation sums of coefficients of Maaß forms in the level aspect. More precisely, letting Π be a Maaß form for $SL_4({{\mathbb{Z}}})$ and f (resp. g) a holomorphic newform of prime level p (resp. trivial level), we show a strengthened upper bound for the sum $\sum_{m,n\ge 1} \lambda_{g}(n) \lambda_{\Pi}(m,1,1) \lambda_{f}(m+pn) \,U{\left({m}/{X}\right) }V{\left ({n}/{H}\right)} $. More generally, for any complex sequence $\lambda_\star (n)\in {{\mathbb{C}}}$, we further obtain a ready-to-use quantitative estimate for the triple correlation sum $\sum_{m,n\ge 1} \lambda_\star (n) \lambda_{\Pi}(m,1,1)\lambda_{f}(m+pn) \,U{\left({m}/{X}\right) }V{\left ({n}/{H}\right)} $. As a result, we present that this type of sum is non-trivial in the regime $p\le X^{1-\varepsilon}$ for any ɛ > 0; notably, there exists a family of newforms such that the triple sum is non-trivial, as long as $H\ge X^{\varepsilon}$. In addition, we obtain an upper bound for the sum of the shape $\sum_{m,n\ge 1}a(m) b(n) \lambda_{f}(m+pn) \,U{\left({m}/{X}\right) }V{\left ({n}/{H}\right)}$ for any complex sequences $\mathbf{a}=\{a(m)\}$ and $\mathbf{b}=\{b(n)\}$. As an application, we consider certain triple correlation sums corresponding to the configuration $GL_M \times GL_N\times GL_2$, $M,N\ge3$ and attain the significant cancellations in these sums.