The $C_2$–spectrum $\mathrm{Tmf}_1(3)$ and its invertible modules

Abstract

We explore the $C_2$–equivariant spectra ${Tmf}_1\left(3\right)$ and ${TMF}_1\left(3\right)$. In particular, we compute their $C_2$–equivariant Picard groups and the $C_2$–equivariant Anderson dual of ${Tmf}_1\left(3\right)$. This implies corresponding results for the fixed-point spectra ${TMF}_0\left(3\right)$ and ${Tmf}_0\left(3\right)$. Furthermore, we prove a real Landweber exact functor theorem.