Nonvanishing of products in $v_2$-periodic families at the prime $3$

Many products amongst $v_2$-periodic families in the stable homotopy groups of spheres are shown not to vanish and some Toda brackets are shown not to contain zero. This is done by carefully studying the action of Adams operations on topological modular forms. A crucial ingredient is Pstragowski’s category of synthetic spectra which affords us the necessary freedom to work with (modified) Adams–Novikov spectral sequences.