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 of the form H\G/K, where G is a non-compact semisimple Lie group, K the maximal compact subgroup of G, and H a discrete subgroup of G. If we take G = SL(2,R), then spaces of this form are precisely hyperbolic surfaces and hyperbolic 2-orbifolds. Automorphic forms then come in two types: modular forms, and eigenfunctions of the hyperbolic Laplacian, also known as Maass forms. 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 all 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 upcoming work with P. Kravchuk and S. Pal.
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