Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch’s trace formula, following Zelditch’s observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $⟨\phi ,{\phi }^2⟩$ uniformly in $t_{\phi }\to \mathrm{\infty }$ and $t_{\phi }\to \mathrm{\infty }$, where $t_{\phi }$ and $t_{\phi }$ are the eigenparameters corresponding to Maass forms $\phi$ and $\phi$ on $X$.