Automorphic forms and cohomology theories on Shimura curves of small discriminant

TL;DR

This paper computes the homotopy groups of spectra linked to Shimura curves with small discriminants, revealing that a generalized truncated Brown-Peterson spectrum is an E_ fty ring spectrum at prime 3, advancing understanding in algebraic topology.

Contribution

It provides explicit computations of automorphic forms and homotopy groups for specific Shimura curves, establishing the E_ fty structure of a generalized BP<2> spectrum at prime 3.

Findings

Homotopy groups of spectra for Shimura curves of discriminants 6, 10, and 14 computed.

Integral rings of automorphic forms on these curves determined.

BP<2> spectrum shown to be an E_ fty ring spectrum at prime 3.

Abstract

We compute the homotopy groups of spectra associated by a theorem of Lurie to the Shimura curves of discriminants 6, 10, and 14, beginning with a computation of integral rings of automorphic forms on these curves. As an application, we find that a generalized truncated Brown-Peterson spectrum BP<2> is an E_\infty ring spectrum at the prime 3.