# Computation of the homotopy of the spectrum tmf

### Abstract

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 The result of this computation is originally due to Hopkins and Mahowald (unpublished).

Publication
Geometry and Topology Monographs 13: Groups, homotopy and configuration spaces (Tokyo 2005), 2008, pp. 11–40
Date