THE 7-CONNECTED COBORDISM RING AT P=3

In this paper, we study the cobordism spectrum M O[8] at the prime 3. This spectrum is important because it is conjectured to play the role for elliptic cohomology that Spin cobordism plays for real K-theory. W e show that the torsion is all killed by 3, and that the Adams-Novikov spectral sequence collapses after only 2 differentials. Many of our m ethods apply more generally.