The K(1)-localized homotopy ring of tmf and tmf is computed at primes 2, 3, and 5, and the resulting tmf-based Adams spectral sequence for the sphere collapses to two lines.
Hopkins, K(1)-local E∞ -ring spectra, Topological modular forms, 2014, pp
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On the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$
The K(1)-localized homotopy ring of tmf and tmf is computed at primes 2, 3, and 5, and the resulting tmf-based Adams spectral sequence for the sphere collapses to two lines.