I will review an analogy between conformal field theory in d dimensions, and spectral theory of automorphic forms on hyperbolic (d+1)-manifolds. Building on this analogy, I will prove a new subconvex bound on triple product L-functions, using the conformal bootstrap. Finally, I will review a theorem of Adve that the bootstrap method knows all there is to know about the spectra and trilinear periods of compact hyperbolic surfaces. Based on works with A. Adve, J. Bonifacio, P. Kravchuk, S. Pal, A. Radcliffe, and G. Rogelberg.
