Motivic invariants of p-adic fields

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 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 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.

💡 Research Summary

This paper undertakes a comprehensive study of the motivic Adams spectral sequences (MASS) that converge to the bigraded homotopy groups of the 2‑completed algebraic Johnson‑Wilson spectra BPGL⟨n⟩ over p‑adic fields. After setting up the motivic stable homotopy category SH(F) for a p‑adic field F (with p≠2 for simplicity), the authors define BPGL⟨n⟩ as the motivic analogue of the classical Johnson‑Wilson spectrum E(n) and then work with its 2‑completion, denoted BPGL⟨n⟩̂₂.

The analysis begins with the base cases n=0 and n=1. For n=0, BPGL⟨0⟩̂₂ coincides with 2‑completed motivic cohomology Hℤ̂₂, and the MASS reduces to the well‑known Ext‑groups over the motivic Steenrod algebra A with ℤ/2 coefficients. The authors employ a ρ‑Bockstein spectral sequence to eliminate the ρ‑torsion that is characteristic of motivic cohomology over p‑adic bases, thereby obtaining a clean E₂‑page. For n=1, BPGL⟨1⟩̂₂ is identified with the connective algebraic K‑theory spectrum kgl̂₂. Here a v₁‑periodicity phenomenon appears: a self‑map induced by the Bott element survives 2‑completion, and the MASS shows that BPGL⟨1⟩̂₂ splits as a free BPGL⟨0⟩̂₂‑module.

The core of the paper treats arbitrary n≥2. The authors combine the ρ‑Bockstein spectral sequence with a May‑type filtration to compute the motivic cohomology of BPGL⟨n⟩̂₂. They demonstrate that the cohomology is built from that of BPGL⟨0⟩̂₂ by successive actions of the motivic periodicity operators v₁, v₂, …, vₙ. Crucially, all differentials involving ρ‑torsion vanish, and the resulting Ext‑groups exhibit a remarkably simple pattern. This leads to the principal splitting theorem:

 BPGL⟨n⟩̂₂ ≅ ⊕_{i=0}^{n} Σ^{2i,i} BPGL⟨0⟩̂₂,

where Σ^{2i,i} denotes a motivic suspension by (2i,i). In other words, the 2‑completed Johnson‑Wilson spectrum over a p‑adic field decomposes as a direct sum of shifted copies of the 2‑completed motivic cohomology spectrum.

An immediate corollary is that the slice spectral sequence for BPGL⟨n⟩̂₂ collapses at the E₁‑page: each summand occupies a distinct slice, and no non‑trivial slice differentials can occur. This “slice collapse” phenomenon is shown to hold uniformly for all n, providing a stark contrast to the more intricate slice behavior over fields such as ℝ or ℂ.

The paper then explores the implications for the motivic Adams‑Novikov spectral sequence (MANSS). Because BPGL⟨n⟩̂₂ splits as a BPGL⟨0⟩̂₂‑module, the MANSS E₂‑page inherits a similarly simple module structure, and higher vₙ‑periodic phenomena that complicate the classical ANSS are absent after 2‑completion. This simplification paves the way for explicit calculations of motivic stable homotopy groups over p‑adic fields and suggests that many of the deep chromatic phenomena in the classical setting become tractable in the motivic p‑adic context.

In conclusion, the authors provide the first systematic description of 2‑completed algebraic Johnson‑Wilson spectra over p‑adic fields, establish their complete splitting over BPGL⟨0⟩̂₂, and prove the collapse of the associated slice spectral sequences. These results lay the groundwork for the sequel paper, which will develop the motivic Adams‑Novikov spectral sequence in this setting and investigate applications to algebraic K‑theory, motivic cobordism, and higher chromatic phenomena in the motivic world.