Strong convergence in the motivic Adams spectral sequence

We prove strong convergence results for the motivic Adams spectral sequence of the sphere spectrum over fields with finite virtual cohomological dimension at the prime 2, and over arbitrary fields at odd primes. We show that the motivic Adams spectral sequence is not strongly convergent over number fields. As applications we give bounds on the exponents of the $(\ell,\eta)$-completed motivic stable stems, and calculate the zeroth $(\ell,\eta)$-completed motivic stable stems.