I will report on joint work with Bobkova, Lachmann, Li, Lima, and Zhang, in which we bound the descent filtration of the exotic Picard group κn, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary β-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the K(n)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at 3n2+1 on the E2n2+2-page.
