The slice spectral sequence for a motivic analogue of the connective $K(1)$-local sphere

TL;DR

This paper computes the slice spectral sequence for a motivic analogue of the connective $K(1)$-local sphere, providing new insights into motivic stable homotopy groups over various fields.

Contribution

It introduces new methods to compute the slice spectral sequence for motivic spectra, including explicit descriptions of differentials and coefficients over different base fields.

Findings

Computed the slice spectral sequence for the motivic $K(1)$-local sphere.

Described $d_1$-differentials using motivic Steenrod operations.

Explicitly described coefficients of motivic Eilenberg--MacLane spectra.

Abstract

We compute the slice spectral sequence for the motivic stable homotopy groups of $L$, a motivic analogue of the connective $K(1)$-local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic $K(1)$-local sphere over arbitrary base fields of characteristic not two. To compute the slice spectral sequence, we prove several results which may be of independent interest. We describe the $d_1$-differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, R{\"o}ndigs, and {\O}stv{\ae}r for the very effective cover of Hermitian K-theory. We also explicitly describe the coefficients of certain motivic Eilenberg--MacLane spectra and compute the slice spectral sequence for the very effective cover of Hermitian K-theory over prime fields.