We show that much of local class theory can be deduced from the Dieudonn\'e-Manin structure theory for $F$-isocrystals on an algebraically closed field of characteristic $p>0$. As a consequence we get a new proof of a formula of Dwork for the norm residue symbol, as well as a "constructive" proof of the local Shafarevich-Weil theorem. This last answers a question of Morava.
Categories: math.NT math.AG