Shifted convolution sums and subconvexity bounds for automorphic L-functions

Abstract

Let f(z)=∑a(m)m(k−1)/2e(mz)∈Sk(N,χ1) be a primitive cusp form of integral weight k, level N, and character χ1. For a smooth weight function g with compact support in [M1,2M1]×[M2,2M2] and positive integers l1, l2, h, the bound ∑l1m1−l2m2=ha(m1)a(m2)¯g(m1,m2)1≪ε(l1M1+l2M2)1/2+θ+ε with θ = 7/64 is shown. As an application, the shifted sum ∑m≤Ma(m)a(m+h)¯ is bounded nontrivially for h≪M64/39−ε⁠. Furthermore, the subconvexity bound Lf(1/2+it,χ)≪εD71/167+ε for the L-function attached to the twist of f with a primitive character χ to modulus D is obtained.