A higher chromatic analogue of the image of J

We prove a higher chromatic analogue of Snaith’s theorem which identifies the K –theory spectrum as the localisation of the suspension spectrum of CP¹ away from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of CP¹ D K.Z; 2/. Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the K.n/–local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross–Hopkins dual of the sphere. Our main technical tool is a K.n/–local notion generalising complex orientation to higher Eilenberg–MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.