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We calculate the group $\kappa_2$ of exotic elements in the $K(2)$-localPicard group at the prime $2$ and find it is a group of order $2^9$ isomorphicto $(\mathbb{Z}/8)^2 \times (\mathbb{Z}/2)^3$. In order to do this we mustdefine and exploit a variety of different ways of constructing elements in thePicard group, and this requires a significant exploration of the theory. Themost innovative technique, which so far has worked best at the prime $2$, isthe use of a $J$-homomorphism from the group of real representations of finitequotients of the Morava stabilizer group to the $K(n)$-local Picard group.

The Exotic $K(2)$-Local Picard Group at the Prime $2$