THE SHIMURA CURVE OF DISCRIMINANT 15 AND TOPOLOGICAL AUTOMORPHIC FORMS

Abstract

We find defining equations for the Shimura curve of discriminant 15 over $\mathbb{Z}[1/15]$. 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.

Keywords

• 55P42 (primary)
• 11F23
• 11G18
• 14G35
• 55P43 (secondary)