Analytic twists of GL2×GL2$\rm GL_2\times \rm GL_2$ automorphic forms

Abstract

Let 𝑓 and 𝑔 be holomorphic or Maass cusp forms for SL 2 (ℤ) with normalized Fourier coefficients 𝜆 𝑓 (𝑛) and 𝜆 𝑔 (𝑛), respectively. In this paper, we prove nontrivial estimates for the sumwhere 𝑒(𝑥) = e 2𝜋𝑖𝑥 , 𝑉(𝑥) ∈  ∞ 𝑐 (1, 2), 𝑡 ≥ 1 is a large parameter and 𝜑(𝑥) is some nonlinear real-valued smooth function. Applications of these estimates include a subconvex bound for the Rankin-Selberg 𝐿-function 𝐿(𝑠, 𝑓 ⊗ 𝑔) in the 𝑡-aspect, an improved estimate for a nonlinear exponential twisted sum and the following asymptotic formula for the sum of the Fourier coefficients of certain GL 5