Shifted convolution sums and subconvexity bounds for automorphic L-functions

Let $f{(z)}=\sum a{(m)}m^{{(k-1)}/2}e{(mz)}\in S_k{(N,\chi_1)}$ be a primitive cusp form of integral weight k, level N, and character χ1. For a smooth weight function g with compact support in ${\lbrack M_1,2M_1\rbrack}\times{\lbrack M_2,2M_2\rbrack}$ and positive integers l1, l2, h, the bound $\sum\nolimits_{l_1m_1-l_2m_2=h}a{(m_1)}\overline{a{(m_2)}}g{(m_1,m_2)}1\ll_\varepsilon{(l_1M_1+l_2M_2)}^{1/2+\theta+\varepsilon}$ with θ = 7/64 is shown. As an application, the shifted sum $\sum\nolimits_{m\leq M}a{(m)}\overline{a{(m+h)}}$ is bounded nontrivially for $h\ll M^{64/39-\varepsilon}$ . Furthermore, the subconvexity bound $L_f{(1/2+it,\chi)}\ll_\varepsilon D^{71/167+\varepsilon}$ for the L-function attached to the twist of f with a primitive character χ to modulus D is obtained.