I will explain how the conformal bootstrap can be adapted to place rigorous bounds on the spectra of automorphic forms on locally symmetric spaces. A locally symmetric space is a space of the form H\G/K, where G is a non-compact semisimple Lie group, K is the maximal compact subgroup of G, and H is a lattice in G. If we take G = SL(2,R), then spaces of this form are precisely hyperbolic surfaces and hyperbolic 2-orbifolds. The bootstrap constraints arise from the associativity of function multiplication on the space H\G, and are very similar to the usual correlator bootstrap equations with G playing the role of the conformal group. For G=SL(2,R), I will use this method to prove upper bounds on the lowest positive eigenvalue of the Laplacian on closed hyperbolic surfaces of a fixed genus. The bounds at genus 2 and genus 3 are very nearly saturated by the Bolza surface and the Klein quartic. This is based on work with P. Kravchuk and S. Pal.