The Picard group is an important invariant of a symmetric monoidal category. In the homotopy category of spectra, these are precisely the isomorphism classes of the n-spheres and the Picard group is a copy of the integers. However, after K(n)-localization, the Picard group can become much more complicated. The K(n)-local categories thus provide examples of interesting Picard groups. Their importance in chromatic homotopy theory is highlighted by the fact that the dualizing object for Brown-Commenetz duality comes from an invertible element.

The K(n)-local Picard groups have been computed at all primes when n=1 and all odd primes when n=2. Mahowald predicted that the torsion in the K(2)-local Picard group at the prime 2 would be very large compared to the torsion in the K(2)-local Picard groups at odd primes. In this talk, I will explain why he was right and explain our current, although incomplete, understanding of the structure of this group.