On the Bousfield-Kan spectral sequence for $Q(2)_{(3)}$

TL;DR

This paper computes the E2-term of the Bousfield-Kan spectral sequence for the spectrum Q(2) localized at 3, providing insights into its homotopy groups and connecting to the structure of the K(2)-local sphere.

Contribution

It offers the first detailed calculation of the E2-term for the spectral sequence converging to Q(2)_{(3)}'s homotopy groups, using advanced techniques from the Adams-Novikov spectral sequence.

Findings

Computed the E2-term of the spectral sequence for Q(2)_{(3)}.

Connected the spectrum's localization to the K(2)-local sphere at prime 3.

Provided new computational tools for understanding homotopy groups of related spectra.

Abstract

We compute the $E_2$-term of the Bousfield-Kan spectral sequence converging to the homotopy groups of the semi-cosimplicial $E_{\infty}$ ring spectrum $Q(2)_{(3)}$. This 3-local spectrum was constructed by M. Behrens using degree 2 isogenies of elliptic curves, and its localization with respect to the 2nd Morava $K$-theory $K(2)$ is "one half" of the $K(2)$-local sphere at the prime 3. The computation in this paper uses techniques developed in the author's previous work on the Adams-Novikov spectral sequence for $Q(2)_{(3)}$, and provides another gateway to the homotopy ring $\pi_*Q(2)_{(3)}$.