Weyl bound for trilinear periods via conformal bootstrap

Let f₁,f₂ be holomorphic modular forms of the same weight for a cocompact lattice Γ< PSL₂(R). We estimate the rate of decay of the coefficients in the expansion of f₁f₂̄ in a Laplace eigenbasis. By specializing our main theorem to the case where $Γ$ 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.