  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 $5$.
