The $C_2$-spectrum $Tmf_1(3)$ and its invertible modules

TL;DR

This paper investigates the structure of certain $C_2$-equivariant spectra related to modular forms, computing their Picard groups, Anderson duals, and establishing a Real Landweber exact functor theorem, advancing understanding in equivariant stable homotopy theory.

Contribution

It provides explicit computations of Picard groups and Anderson duals for $Tmf_1(3)$, and proves a Real Landweber exact functor theorem, offering new tools for equivariant spectra analysis.

Findings

Computed $C_2$-equivariant Picard groups of $Tmf_1(3)$ and $TMF_1(3)$

Determined the $C_2$-equivariant Anderson dual of $Tmf_1(3)$

Established a Real Landweber exact functor theorem

Abstract

We explore the $C_2$-equivariant spectra $Tmf_1(3)$ and $TMF_1(3)$. In particular, we compute their $C_2$-equivariant Picard groups and the $C_2$-equivariant Anderson dual of $Tmf_1(3)$. This implies corresponding results for the fixed point spectra $TMF_0(3)$ and $Tmf_0(3)$. Furthermore, we prove a Real Landweber exact functor theorem.