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

In this paper, 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. The same analysis also allows us to express the exotic Picard group of $K(n)$-local modules over the homotopy fixed points spectrum EₙhN, where N is the normalizer in Gₙ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group H²ⁿ⁺¹(N,π₂ₙEₙ).