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

## TL;DR

This paper computes the $K(1)$-local homotopy groups of the spectrum $ ext{tmf} igwedge ext{tmf}$ to advance understanding of the $ ext{tmf}$-based Adams spectral sequence, utilizing Hopkins' presentation of $L_{K(1)} ext{tmf}$.

## Contribution

It provides the first explicit computation of the $K(1)$-local homotopy of $ ext{tmf} igwedge ext{tmf}$ and describes the associated Adams spectral sequence.

## Key findings

- Computed the $K(1)$-local homotopy groups of $ ext{tmf} igwedge ext{tmf}$.
- Described the $K(1)$-local $ ext{tmf}$-based Adams spectral sequence.
- Utilized Hopkins' presentation of $L_{K(1)} ext{tmf}$.

## Abstract

As a step towards understanding the $\mathrm{tmf}$-based Adams spectral sequence, we compute the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$, using a small presentation of $L_{K(1)}\mathrm{tmf}$ due to Hopkins. We also describe the $K(1)$-local $\mathrm{tmf}$-based Adams spectral sequence.