Revisiting the ${\beta }_1$-action on the $3$-primary stable homotopy groups of spheres

Let ${\beta }_1$ be the first $3$-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that ${\beta }_1^5\mathrm{\ne }0$, whilst ${\beta }_1^6=0$. Shimomura generalised this to the $144$-periodic family generated by ${\beta }_1$, written as $\{{\beta }_{1+9s}{\}}_{s\ge 0}$, and showed that any $5$-fold product ${\prod }_5{\beta }_{1+9s}\mathrm{\ne }0$, whilst all $6$-fold products ${\prod }_6{\beta }_{1+9s}=0$. In this article, we give a simple proof of these results as well as some generalisations to other $144$-periodic families. Our tools include BP-synthetic spectra, and the well-known Adams–Novikov spectral sequence for the spectrum of topological modular forms at the prime $3$ as well as its Adams operations.