Picard groups of higher real \(K\)-theory spectra at height \(p-1\)

Abstract

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real \(K\)-theory spectra of Hopkins and Miller at height \(n=p-1\), for \(p\) an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra \(E_n^{hG}\), where \(E_n\) is Lubin-Tate \(E\)-theory at the prime \(p\) and height \(n=p-1\), and \(G\) is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.