Automorphic forms and cohomology theories on Shimura curves of small discriminant

Abstract

We apply Lurie's theorem to produce spectra associated to 1-dimensional formal group laws on the Shimura curves of discriminants 6, 10, and 14. We compute rings of automorphic forms on these curves and the homotopy of the associated spectra. At p = 3 , we find that the curve of discriminant 10 recovers much the same as the topological modular forms spectrum, and the curve of discriminant 14 gives rise to a model of a truncated Brown–Peterson spectrum as an E ∞ ring spectrum.