Motivic Brown-Peterson invariants of the rationals

TL;DR

This paper computes the motivic Adams spectral sequence for Brown-Peterson spectra over rationals, introduces a motivic Hasse principle, and connects these results to classical K-theory theorems.

Contribution

It provides the first computation of the homotopy groups of BP<n> over 2-adic rationals and establishes a motivic Hasse principle for these spectra.

Findings

Computed the motivic Adams spectral sequence for BP<n> over Q

Proved a motivic Hasse principle for BP<n> spectra

Derived classical K-theory results from motivic computations

Abstract

Fix the base field Q of rational numbers and let BP<n> denote the family of motivic truncated Brown-Peterson spectra over Q. We employ a "local-to-global" philosophy in order to compute the motivic Adams spectral sequence converging to the bi-graded homotopy groups of BP<n>. Along the way, we provide a new computation of the homotopy groups of BP<n> over the 2-adic rationals, prove a motivic Hasse principle for the spectra BP<n>, and deduce several classical and recent theorems about the K-theory of particular fields.