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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 $p-1$ doesnot divide $h$. The main idea is to use the Gross-Hopkins duality to relateboth questions to certain Greek letter element computations in chromatichomotopy theory. Classical results of Miller-Ravenel-Wilson then imply that anexotic 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 holdsmodulo the invariant prime ideal $I_{h-1}$. We further show that this specialcase of the Vanishing Conjecture implies the exotic Picard group $\kappa_h$ iszero at height $3$ and prime $5$. Both results can be thought of as a firststep towards proving the vanishing of $\kappa_3$ at prime $5$.

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