\(p\)-adic solubility in families

Daniel Loughran (Manchester)

Frank Adams 1,

In 1965 Ax and Kochen proved a famous theorem concerning the \(p\)-adic solubility of homogeneous polynomials of small degree.  This theorem was originally proved using tools from model theory, however Denef, following a strategy suggested by Colliot-Thélène, recently found a purely geometric proof that moreover gives results concerning \(p\)-adic solubility in greater generality.

In this talk we build upon Denef's work and give a criterion that completely classifies those families of varieties for which an analogue of the Ax-Kochen theorem holds. This work is joint with Arne Smeets and Alexei Skorobogatov.
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