Stable Adams operations on $RO(C_2)$-graded homotopy groups

TL;DR

This paper computes the stable Adams operations on the $RO(C_2)$-graded homotopy groups of specific $C_2$-spectra, namely Real $K$-theory and topological modular forms of level 3, advancing understanding of their equivariant structures.

Contribution

It provides explicit calculations of Adams operations on the homotopy groups of $ extbf{KR}$ and $ extbf{TMF}_1(3)$, which were previously not known.

Findings

Explicit formulas for Adams operations on $ extbf{KR}$ homotopy groups.

Explicit formulas for Adams operations on $ extbf{TMF}_1(3)$ homotopy groups.

Enhanced understanding of equivariant homotopy groups of these spectra.

Abstract

The $C_2$-spectrum of Atiyah's Real $K$-theory is denoted by $\mathbf{KR}$ and the $C_2$-spectrum of topological modular forms of level structure $\Gamma_1(3)$ by $\mathbf{TMF}_1(3)$. In this short note we compute the $C_2$-equivariant stable Adams operations on the $RO(C_2)$-graded homotopy groups of $\mathbf{KR}$ and $\mathbf{TMF}_1(3)$.