Motivic invariants of p-adic fields

TL;DR

This paper analyzes the motivic Adams spectral sequences for 2-complete algebraic Johnson-Wilson spectra over p-adic fields, revealing their structure and splitting properties, and sets the stage for further studies of motivic invariants in this context.

Contribution

It provides a complete analysis of the motivic Adams spectral sequences for BPGL<n> over p-adic fields and shows the splitting and collapse of the spectral sequence, advancing understanding of motivic invariants.

Findings

Spectral sequences for BPGL<n> over p-adic fields are fully analyzed.

BPGL<n> spectra split over BPGL<0> in this setting.

The slice spectral sequence for BPGL collapses over p-adic fields.

Abstract

We provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectra BPGL<n> over p-adic fields. These spectra interpolate between integral motivic cohomology (n=0), a connective version of algebraic K-theory (n=1), and the algebraic Brown-Peterson spectrum. We deduce that, over p-adic fields, the 2-complete BPGL<n> split over 2-complete BPGL<0>, implying that the slice spectral sequence for BPGL collapses. This is the first in a series of two papers investigating motivic invariants of p-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.