Motivic Brown–Peterson invariants of the rationals

Abstract

Let $\mathsf{BP}\langle n\rangle$, $0\le n\le \infty$, denote the family of motivic truncated Brown–Peterson spectra over $\mathbb{Q}$. We employ a “local-to-global” philosophy in order to compute the bigraded homotopy groups of $\mathsf{BP}\langle n\rangle$. Along the way, we produce a computation of the homotopy groups of $\mathsf{BP}\langle n\rangle$ over ${\mathbb{Q}}_2$, prove a motivic Hasse principle for the spectra $\mathsf{BP}\langle n\rangle$, and reprove several classical and recent theorems about the $K$–theory of particular fields in a streamlined fashion. We also compute the bigraded homotopy groups of the 2–complete algebraic cobordism spectrum $\mathsf{MGL}$ over $\mathbb{Q}$.