  This paper contains a complete computation of the homotopy ring of the
spectrum of topological modular forms constructed by Hopkins and Miller. The
computation is done away from 6, and at the (interesting) primes 2 and 3
separately, and in each of the latter two cases, a sequence of algebraic
Bockstein spectral sequences is used to compute the E_2 term of the elliptic
Adams-Novikov spectral sequence from the elliptic curve Hopf algebroid. In a
further step, all the differentials in the latter spectral sequence are
determined. The result of this computation is originally due to Hopkins and
Mahowald (unpublished).
