Abstract Arne Smeets

Arne Smeets will speak on

A geometric Ax-Kochen theorem

Abstract: The celebrated Ax-Kochen theorem states that for every positive integer d, there is a finite set S_d of primes such that if p is a prime not in S_d, then every homogeneous polynomial of degree d over Q_p in more than d^2 variables has a non-trivial zero. This classical result was originally proven using model theory. I will present a geometric statement which generalizes the Ax-Kochen theorem and which is optimal. This uses some birational and toroidal geometry. This is joint work with Dan Loughran and Alexei Skorobogatov, building on earlier work of Colliot-Thélène and Denef.