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.