RO(G)-Graded Homotopy Fixed Point Spectral Sequence for Height 2 Morava E-Theory

Abstract

We consider $G=Q_8,\textrm{SD}_{16},G_{24},$ and $G_{48}$ as finite subgroups of the Morava stabilizer group which acts on the height 2 Morava E-theory ${\textbf{E}}_2$ at the prime 2. We completely compute the G-homotopy fixed point spectral sequences of ${\textbf{E}}_2$. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the $(*-\sigma _i)$-graded $Q_8$- and $\textrm{SD}_{16}$-homotopy fixed point spectral sequences, where $\sigma _i$ is a non-trivial one-dimensional representation of $Q_8$.

Funding

Shanghai Science and Technology Development Foundation(No. 20QA1401600)

National Science Foundation(No. DMS-1926686)