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 $\left(n,0\right)$ motivic stable homotopy group of spheres over the finite field with $q$ elements, ${\mathbb{F}}_q$. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems ${\pi }_{n,0}{\left({\mathbb{F}}_q\right)}_2^{\wedge }$ for $0\le n\le 18$ for all finite fields and ${\pi }_{19,0}{\left({\mathbb{F}}_q\right)}_2^{\wedge }$ and ${\pi }_{20,0}{\left({\mathbb{F}}_q\right)}_2^{\wedge }$ when $q\equiv 1mod4$ assuming Morel’s connectivity theorem for ${\mathbb{F}}_q$ holds.