Weyl bound for trilinear periods via conformal bootstrap

Let $f_1,f_2$ be holomorphic modular forms of the same weight for a cocompact lattice $\mathrm{\Gamma }<{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbf{R})$. We estimate the rate of decay of the coefficients in the expansion of $f_1\overline{f_2}$ in a Laplace eigenbasis. By specializing our main theorem to the case where $\mathrm{\Gamma }$ is arithmetic, we obtain new instances of the Weyl bound for triple product $L$-functions in the spectral aspect. Our method builds on the conformal bootstrap in physics.