Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality

In this paper, we study the exotic $K(h)$-local Picard groups $\kappa_h$ when $2p-1=h^2$ and the homological Chromatic Vanishing Conjecture when $p-1$ does not divide $h$. The main idea is to use the Gross-Hopkins duality to relate both questions to certain Greek letter element computations in chromatic homotopy theory. Classical results of Miller-Ravenel-Wilson then imply that an exotic element at height $3$ and prime $5$ is not detected by the type-$2$ complex $V(1)$. For the homological Vanishing Conjecture, we prove it holds modulo the invariant prime ideal $I_{h-1}$. We further show that this special case of the Vanishing Conjecture implies the exotic Picard group $\kappa_h$ is zero at height $3$ and prime $5$. Both results can be thought of as a first step towards proving the vanishing of $\kappa_3$ at prime $5$.