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As 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.

On the K(1)-local homotopy of $$\mathrm {tmf}\wedge \mathrm {tmf}$$