The motivic lambda algebra and motivic Hopf invariant one problem

Abstract: We investigate forms of the Hopf invariant one problem in motivic homotopy theory over arbitrary base fields of characteristic not equal to $2$. Maps of Hopf invariant one classically arise from unital products on spheres, and one consequence of our work is a classification of motivic spheres represented by smooth schemes admitting a unital product. The classical Hopf invariant one problem was resolved by Adams, following his introduction of the Adams spectral sequence. We introduce the motivic lambda algebra as a tool to carry out systematic computations in the motivic Adams spectral sequence. Using this, we compute the $E_2$-page of the $\mathbb{R}$-motivic Adams spectral sequence in filtrations $f \leq 3$. This universal case gives information over arbitrary base fields. We then study the $1$-line of the motivic Adams spectral sequence. We produce differentials $d_2(h_{a+1}) = (h_0+\rho h_1)h_a^2$ over arbitrary base fields, which are motivic analogues of Adams' classical differentials. Unlike the classical case, the story does not end here, as the motivic $1$-line is significantly richer than the classical $1$-line. We determine all permanent cycles on the $\mathbb{R}$-motivic $1$-line, and explicitly compute differentials in the universal cases of the prime fields $\mathbb{F}_q$ and $\mathbb{Q}$, as well as $\mathbb{Q}_p$ and $\mathbb{R}$.