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, which states that the $p$-adics are usually $C_{2}$ fields. The method of proof utilizes a construction from mathematical logic called the ultraproduct.
Citation
Bradley Duesler. Amanda Knecht. "Rationally connected varieties over the maximally unramified extension of $p$-adic fields." Rocky Mountain J. Math. 47 (8) 2605 - 2617, 2017. https://doi.org/10.1216/RMJ-2017-47-8-2605
Information