Two-complete stable motivic stems over finite fields

Abstract

Let $\ell$ be a prime and $q = p^{\nu}$ where $p$ is a prime different from $\ell$. We show that the $\ell$-completion of the $n$th stable homotopy group of spheres is a summand of the $\ell$-completion of the $(n, 0)$ motivic stable homotopy group of spheres over the finite field with $q$ elements $F_q$. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems $\pi_{n, 0}(F_q)^{\wedge}_2$ for $0\leq n\leq 18$. Additionally, we compute $\pi_{19, 0}(F_q)^{\wedge}_2$ and $\pi_{20, 0}(F_q)^{\wedge}_2$ when $q \equiv 1 \bmod 4$ assuming Morel's connectivity theorem for $F_q$ holds.Comment: 4 figures, 2 tables. Published version now availabl

The Chow $t$-structure on the $\infty$-category of motivic spectra

“…• Over finite fields for the spectrum ½ ∧ 2 , this was computed in [WØ17] by a long exact sequence with the other two terms the C-motivic Ext. The boundary homomorphism corresponds to the Steenrod operation action on powers of τ .…”

“…As an example, we can show that over finite fields, our method gives proofs of previously undetermined Adams differentials in [WØ17] (See Remark 5.15 for more details). 1.7.…”

Strong convergence in the motivic Adams spectral sequence

“…We use a Bockstein spectral sequence to extend Adams's isomorphism (4) to general fields. See [Hil11, 2.2] or [Wil16,p. 51] for Bockstein spectral sequences at the prime 2 or at odd primes over finite fields.…”

The eta-inverted sphere over the rationals

“…We now identify some classes in π * * (½ ∧ 2 )(F q ) for finite fields F q using the analysis of the motivic Adams spectral sequence by Wilson and Østvaer in [WØ17]. Over a finite field F q with q ≡ 1 mod 4, define ε ∈ π 8,5 (½ ∧ 2 ) ∼ = (Z/2) 4 to be a class detected by c 0 .…”