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 ${\pi }_{n,0}(F_q{)}_2^{\wedge }$ for $0\le n\le 18$. Additionally, we compute ${\pi }_{19,0}(F_q{)}_2^{\wedge }$ and ${\pi }_{20,0}(F_q{)}_2^{\wedge }$ when $q\equiv 1\mathrm{mod}4$ assuming Morel's connectivity theorem for $F_q$ holds.

4 figures, 2 tables. Published version now available

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