Submission date: 14 January 2016.

Abstract:

[To appear in a special volume of the American Journal of Mathematics dedicated to the memory of Professor Jun-ichi Igusa]
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.

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Full text arXiv 1601.03607: pdf, ps.