Higher real K-theories and topological automorphic forms

TL;DR

This paper investigates the relationship between higher real K-theories and topological automorphic forms, establishing conditions under which certain K-theories are summands of localized TAF-spectra, with explicit algebraic constructions.

Contribution

It proves that for specific primes and subgroup conditions, the higher real K-theory EO_n is a summand of a localized TAF-spectrum, and provides explicit algebraic embeddings in other cases.

Findings

EO_n is a summand of K(n)-localization of TAF-spectrum for p in {2,3,5,7} and specific subgroup conditions.

Explicit presentation of a global division algebra with involution embedding the subgroup G.

Negative results for other odd prime cases regarding the summand relationship.

Abstract

Given a maximal finite subgroup G of the nth Morava stabilizer group at a prime p, we address the question: is the associated higher real K-theory EO_n a summand of the K(n)-localization of a TAF-spectrum associated to a unitary similitude group of type U(1,n-1)? We answer this question in the affirmative for p in {2, 3, 5, 7} and n = (p-1)p^{r-1} for a maximal finite subgroup containing an element of order p^r. We answer the question in the negative for all other odd primary cases. In all odd primary cases, we to give an explicit presentation of a global division algebra with involution in which the group G embeds unitarily.