Two-complete stable motivic stems over finite fields

Let ℓ be a prime and q = pυ, where p is a prime different from ℓ. We show that the ℓ–completion of the nth stable homotopy group of spheres is a summand of the ℓ–completion of the (n, 0) motivic stable homotopy group of spheres over the finite field with q elements, Fq. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems πn,0(Fq)^2 for 0 ≤ n ≤ 18 for all finite fields and π19,0(Fq)^2 and π20,0(Fq)^2 when q ≡ 1 mod 4 assuming Morel’s connectivity theorem for Fq holds.