Exotic Motivic Periodicities

One can attempt to study motivic homotopy groups by mimicking the classical (non-motivic) chromatic approach. There are however major differences, which makes the motivic story more complicated and still not well understood. For example, classically the $p$-local sphere spectrum $S^0_{(p)} $ admits an essentially unique non-nilpotent self-map, which is not the case motivically, since Morel showed that the first Hopf map $\eta \colon S^{1,1} \to S^{0,0}$ is non-nilpotent. In the same way that the non-nilpotent self-map $2 = v_0 \in \pi_{\ast,\ast}(S^{0,0})$ starts the usual chromatic story of $v_n$-periodicity, there is a similar theory starting with the non-nilpotent element $\eta \in \pi_{\ast,\ast}(S^{0,0})$, which Andrews-Miller denoted by $\eta = w_0$. In this paper we investigate the beginning of the motivic story of $w_n$-periodicity when the base scheme is $\mathbf{Spec} \! \ \mathbb{C}$. In particular, we construct motivic fields $K(w_n)$ designed to detect such $w_n$-periodic phenomena, in the same way that $K(n)$ detects $v_n$-periodic phenomena. In the hope of detecting motivic nilpotence, we also construct a more global motivic spectrum $wBP$ with homotopy groups $\pi_{\ast,\ast}(wBP) \cong \mathbb{F}_2[w_0, w_1, \ldots]$.