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

We investigate the triple correlation sums of coefficients of Maaß forms in the level aspect. More precisely, letting Π be a Maaß form for SL₄(Z) and f (resp. g) a holomorphic newform of prime level p (resp. trivial level), we show a strengthened upper bound for the sum ∑m,n≥ 1 λg(n) λΠ(m,1,1) λf(m+pn) \,U(m/X) V (n/H). More generally, for any complex sequence λ_ (n)∈ C, we further obtain a ready-to-use quantitative estimate for the triple correlation sum ∑m,n≥ 1 λ_ (n) λΠ(m,1,1)λf(m+pn) \,U(m/X) V (n/H). As a result, we present that this type of sum is non-trivial in the regime p≤ X1-ε for any ɛ > 0; notably, there exists a family of newforms such that the triple sum is non-trivial, as long as H≥ Xε. In addition, we obtain an upper bound for the sum of the shape ∑m,n≥ 1a(m) b(n) λf(m+pn) \,U(m/X) V (n/H) for any complex sequences a=(m)\ and b=(n)\. As an application, we consider certain triple correlation sums corresponding to the configuration GLM × GLN× GL₂, M,N≥3 and attain the significant cancellations in these sums.