The homotopy groups of the L-2-localized Toda-Smith spectrum V(1) at the prime 3

In this paper, we try to compute the homotopy groups of the L-2-localized Toda-Smith spectrum V(1) at the prime 3 by using the Adams-Novikov spectral sequence, and have almost done so. This computation involves non-trivial differentials d(5) and d(9) of the Adams-Novikov spectral sequence, different from the case p > 3. We also determine the homotopy groups of some L-2-localized finite spectra relating to V(1). We further show some of the non-trivial differentials on elements relating so-called beta-elements in the Adams-Novikov spectral sequence for kappa* (S-0).