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

Abstract

In this paper, we study the exotic K(h)-local Picard groups κh when 2p−1=h2 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 Ih−1. We further show that this special case of the Vanishing Conjecture implies the exotic Picard group κh is zero at height 3 and prime 5. Both results can be thought of as a first step towards proving the vanishing of κ3 at prime 5.