Rocky Mountain Journal of Mathematics

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

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.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 8 (2017), 2605-2617.

Dates
First available in Project Euclid: 3 February 2018

https://projecteuclid.org/euclid.rmjm/1517648601

Digital Object Identifier
doi:10.1216/RMJ-2017-47-8-2605

Mathematical Reviews number (MathSciNet)
MR3760308

Zentralblatt MATH identifier
06840990

Subjects
Primary: 14G05: Rational points 14M22: Rationally connected varieties

Citation

Duesler, Bradley; Knecht, Amanda. Rationally connected varieties over the maximally unramified extension of $p$-adic fields. Rocky Mountain J. Math. 47 (2017), no. 8, 2605--2617. doi:10.1216/RMJ-2017-47-8-2605. https://projecteuclid.org/euclid.rmjm/1517648601