Geometric proofs of theorems of Ax-Kochen and Eršov

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-Er{\v{s}}ov transfer principle for local fields, and of quantifier elimination theorems of Basarab and Pas.