Rationally connected varieties over the maximally unramified extension of p-adic fields

TL;DR

This paper proves that rationally connected varieties over the maximally unramified extension of p-adic fields generally have rational points, extending known results from complex and finite fields using ultraproducts and model theory techniques.

Contribution

It extends the existence of rational points on rationally connected varieties to the mixed characteristic case over the maximally unramified extension of p-adic fields, using ultraproduct methods.

Findings

Rationally connected varieties over the maximally unramified extension of p-adics usually have rational points.

The proof employs ultraproducts to lift results from equicharacteristic to mixed characteristic.

The approach connects algebraic geometry with model theory techniques.

Abstract

A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here we show that rationally connected varieties over the maximally unramified extension of the p-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result saying that the p-adics are usually $C_{2}$ fields. The method of proof utilizes a construction from mathematical logic called the ultraproduct. The ultraproduct is used to lift the de Jong, Starr result in the equicharacteristic case to the mixed characteristic case.