  We give an algebraic geometric proof of the Theorem of Ax and Kochen on
p-adic diophantine equations in many variables. Unlike Ax-Kochen's proof, ours
does not use any notions from mathematical logic and is based on weak
toroidalization of morphisms. We also show how this geometric approach yields
new proofs of the Ax-Kochen-Ersov transfer principle for local fields, and of
quantifier elimination theorems of Basarab and Pas.
