The Shimura curve of discriminant 15 and topological automorphic forms

TL;DR

This paper explicitly describes the Shimura curve of discriminant 15, computes its automorphic forms, and applies these results to determine the homotopy groups of related topological automorphic form spectra.

Contribution

It provides explicit equations for the Shimura curve of discriminant 15 and computes the associated automorphic forms and homotopy groups of related spectra.

Findings

Explicit equations for the Shimura curve of discriminant 15

Computed the graded ring of automorphic forms over 2-adic integers

Determined the homotopy groups of topological automorphic form spectra

Abstract

We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of "topological automorphic forms" associated to this curve, as well as one associated to a quotient by an Atkin-Lehner involution.