  We consider states with large angular momentum to facilitate the study of the
M-theory regime of the AdS_4/CFT_3 correspondence. We study the duality between
M-theory in AdS_4xS^7/Z_k and the ABJM N=6 Chern-Simons-matter theory with
gauge group U(N)xU(N) and level k, taking N large and k of order 1. In this
regime the lack of an explicit formulation of M-theory in AdS_4xS^7/Z_k makes
the gravity side difficult, while the CFT is strongly coupled and the planar
approximation is not applicable. To overcome these difficulties, we focus on
states on the gravity side with large angular momentum J>>1 and identify the
dual operators in the CFT, thereby establishing the AdS/CFT dictionary in this
sector. Natural approximation schemes arise on both sides thanks to the
presence of the small parameter 1/J. On the AdS side, we use the matrix model
of M-theory on the maximally supersymmetric pp-wave background with matrices of
size J/k. A perturbative treatment of this matrix model provides a good
approximation to M-theory in AdS_4xS^7/Z_k when N^{1/3}<<J<<N^{1/2}. On the CFT
side, we study the theory on S^2xR with magnetic flux J/k. A Born-Oppenheimer
type expansion arises naturally for large J in spite of the theory being
strongly coupled. The energy spectra on the two sides agree at leading order.
This provides a non-trivial test of the AdS_4/CFT_3 correspondence including
near-BPS observables associated with membrane degrees of freedom, thus
verifying the duality beyond the previously studied sectors corresponding to
either BPS observables or the type IIA string regime.
