Two-complete stable motivic stems over finite fields

Let $\mathrm{\ell }$ be a prime and $q=p^{\nu }$ where $p$ is a prime different from $\mathrm{\ell }$. We show that the $\mathrm{\ell }$-completion of the $n$th stable homotopy group of spheres is a summand of the $\mathrm{\ell }$-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 $0\leq n\leq 18$\mathrm{.}Additionally,wecompute$\pi_{19, 0}(F_q)^{\wedge}2$and$\pi{20, 0}(F_q)^{\wedge}_2$when$q \equiv 1 \bmod 4$ assuming Morel’s connectivity theorem for $F_q$ holds.