Bounding the \(K(p-1)\)-local exotic Picard group at \(p>3\)

Abstract

In this paper, we bound the descent filtration of the exotic Picard group \(\kappa_n\), 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 \(\beta\)-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 \(3n^2+1\) on the \(E_{2n^2+2}\)-page. The same analysis also allows us to express the exotic Picard group of \(K(n)\)-local modules over the homotopy fixed points spectrum \(\mathrm{E}_n^{hN}\), where \(N\) is the normalizer in \(\mathbb{G}_n\) of a finite cyclic subgroup of order \(p\), as a subquotient of a single continuous cohomology group \(H^{2n+1}(N,\pi_{2n}\mathrm{E}_n)\).