The Adams differentials on the classes \(h_{j}^{3}\)

In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams (Ann. Math. (2) 72:20–104 1960) computed the differentials on the classes $h_j$ $h_{j}$ , resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill–Hopkins–Ravenel (Ann. Math. (2) 184(1):1–262 2016) proved that the classes $h_j^2$ $h_{j}^{2}$ support non-trivial differentials for $j\ge 7$ $j \geq 7$ , resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes $h_j^2$ $h_{j}^{2}$ for $j\ge 7$ $j \geq 7$ and the fate of $h_6^2$ $h_{6}^{2}$ remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial $d_4$ $d_{4}$ -differentials on the classes $h_j^3$ $h_{j}^{3}$ for $j\ge 6$ $j \geq 6$ , confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory—ℂ-motivic stable homotopy theory and ${\mathbb{F}}_2$ $\mathbb{F}_{2}$ -synthetic homotopy theory—both in an essential way. Along the way, we also show that $h_j^2$ $h_{j}^{2}$ survives to the Adams $E_5$ $E_{5}$ -page and that $h_6^2$ $h_{6}^{2}$ survives to the Adams $E_9$ $E_{9}$ -page.