AbstractAs a step towards understanding the $$\mathrm {tmf}$$
tmf
-based Adams spectral sequence, we compute the K(1)-local homotopy of $$\mathrm {tmf}\wedge \mathrm {tmf}$$
tmf
∧
tmf
, using a small presentation of $$L_{K(1)}\mathrm {tmf}$$
L
K
(
1
)
tmf
due to Hopkins. We also describe the K(1)-local $$\mathrm {tmf}$$
tmf
-based Adams spectral sequence.
AbstractHecke operators are used to investigate part of the E2-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of Ext1 which combines use of classical Hecke operators and p-adic Hecke operators due to Serre.
The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.