Geometric proofs of theorems of Ax-Kochen and Ersov

2016/01/14 by Denef, Jan
#Algebraic Geometry (math.AG) #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1601.03607

Abstract

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.

Related

• The Ax-Kochen-Ershov Theorem
• The Ax-Kochen Theorem: an application of model theory to algebra
• B(G) for all local and global fields
• An Ax-Kochen-Ershov Theorem for Monotone Differential-Henselian Fields
• p-adic geometry