Motivic Mahowald invariants over general base fields

Abstract

The motivic Mahowald invariant was introduced in [J. D. Quigley, Algebr. Geom. Topol. 19, No. 5, 2485–2534 (2019; Zbl 1436.55016)] and [J. D. Quigley, J. Topol. 14, No. 2, 369–418 (2021; Zbl 07381853)] to study periodicity in the $\mathbb{C}$- and $\mathbb{R}$-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field $F$ of characteristic not two and use it to study periodicity in the $F$-motivic stable stems. In particular, we construct lifts of some of Adams' classical $v_1$-periodic families [J. F. Adams, Topology 5, 21–71 (1966; Zbl 0145.19902)] and identify them as the motivic Mahowald invariants of powers of $2+\rho \eta$.