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Register a new userMotivic invariants of p-adic fields
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Authors
Kyle M. Ormsby
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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.
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We explain how to reconstruct the category of Artin-Tate $mathbb{R}$-motivic spectra as a deformation of the purely topological $C_2$-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of $C_2$-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of $tau$ philosophy that has revolutionized classical stable homotopy theory. A key observation is that the Artin-Tate subcategory of $mathbb{R}$-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the $tau$ map, which is a feature conspicuously absent from the cellular category.
The motivic Mahowald invariant was introduced in cite{Qui19a} and cite{Qui19b} to study periodicity in the $mathbb{C}$- and $mathbb{R}$-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field $F$ of characteristic not two and use it to study periodicity in the $F$-motivic stable stems. In particular, we construct lifts of some of Adams classical $v_1$-periodic families cite{Ada66} and identify them as the motivic Mahowald invariants of powers of $2+rho eta$.
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.
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).
We generalize the Mahowald invariant to the $mathbb{R}$-motivic and $C_2$-equivariant settings. For all $i>0$ with $i equiv 2,3 mod 4$, we show that the $mathbb{R}$-motivic Mahowald invariant of $(2+rho eta)^i in pi_{0,0}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Adams classical $v_1$-periodic families, and for all $i > 0$, we show that the $mathbb{R}$-motivic Mahowald invariant of $eta^i in pi_{i,i}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Andrews $mathbb{C}$-motivic $w_1$-periodic families. We prove analogous results about the $C_2$-equivariant Mahowald invariants of $(2+rho eta)^i in pi_{0,0}^{C_2}(S^{0,0})$ and $eta^i in pi_{i,i}^{C_2}(S^{0,0})$ by leveraging connections between the classical, motivic, and equivariant stable homotopy categories. The infinite families we construct are some of the first periodic families of their kind studied in the $mathbb{R}$-motivic and $C_2$-equivariant settings.
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