Two-complete stable motivic stems over finite fields
Let $\ell$ be a prime and $q = p^ν$ 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 $π_{n, 0}(F_q)^{\wedge}_2$ for $0\leq n\leq 18$. Additionally, we compute $π_{19, 0}(F_q)^{\wedge}_2$ and $π_{20, 0}(F_q)^{\wedge}_2$ when $q \equiv 1 \bmod 4$ assuming Morel's connectivity theorem for $F_q$ holds.
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