Kyle M. Ormsby

On the homotopy of Q(3) and Q(5) at the prime 2

We study modular approximations Q(l), l = 3,5, of the K(2)-local sphere at the prime 2 that arise from l-power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(l) and record Hill, Hopkins, and Ravenels computation of the homotopy groups of TMF_0(5). Using these tools and formulas of Mahowald and Rezk for Q(3) we determine the image of Shimomuras 2-primary divided beta-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use low-dimensional computations of the homotopy of Q(3) and Q(5) to explore the role of these spectra as approximations to the K(2)-local sphere.

Motivic Brown–Peterson invariants of the rationals

Let BPhni, 0 n1 , denote the family of motivic truncated Brown‐Peterson spectra over Q. We employ a “local-to-global” philosophy in order to compute the bigraded homotopy groups of BPhni. Along the way, we produce a computation of the homotopy groups of BPhni over Q2 , prove a motivic Hasse principle for the spectra BPhni, 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 MGL over Q. 55T15; 19D50, 19E15

MOTIVIC INVARIANTS OF p-ADIC FIELDS

We provide a complete analysis of the motivic Adams spec- tral sequences converging to the bigraded coefficients of the 2-complete algebraic Johnson-Wilson spectraBPGLhni overp-adic fields. These spec- tra interpolate between integral motivic cohomology (n = 0), a connec- tive version of algebraic K-theory (n = 1), and the algebraic Brown- Peterson spectrum (n = 1). We deduce that, over p-adic fields, the 2- complete BPGLhni splits over 2-complete BPGLh0i, implying that the slice spectral sequence forBPGL collapses. This is the first in a series of two papers investigating motivic invari- ants ofp-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.

Primes and fields in stable motivic homotopy theory

Let F be a field of characteristic different than 2. We establish surjectivity of Balmers comparison map rho^* from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.