The Adams differentials on the classes h³_j

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 , 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 support non-trivial differentials for , resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes for and the fate of remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial -differentials on the classes for , confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory—ℂ-motivic stable homotopy theory and -synthetic homotopy theory—both in an essential way. Along the way, we also show that survives to the Adams -page and that survives to the Adams -page.