Analytic Twists of GL3 x GL2 Automorphic Forms

Let pi be a Hecke-Maass cusp form for SL3(Z) with normalized Hecke eigenvalues lambda(pi) (n, r). Let f be a holomorphic or Maass cusp form for SL2( Z) with normalized Hecke eigenvalues lambda(f) (n). In this paper, we are concerned with obtaining nontrivial estimates for the sum Sigma(r,n >= 1) lambda(pi) (n,r)lambda(f) (n) e (t phi(r2n/N)) V (r(2)n/N) where e(x) = e(2pix), V(x) is an element of C-c(infinity) (0,infinity), t >= 1 is a large parameter and phi(x) is some realvalued smooth function. As applications, we give an improved subconvexity bound for GL(3) x GL(2) L-functions in the t-aspect and under the Ramanujan-Petersson conjecture we derive the following bound for sums of GL(3) x GL(2) Fourier coefficients Sigma(r2n <= x) lambda(pi) (r,n) lambda(f)(n) << (pi,f,epsilon) x(5/7-1/364+epsilon) for any epsilon > 0, which breaks for the 1st time the barrier O(x(5/7+epsilon)) in a work by Friedlander-Iwaniec.