## Rocky Mountain Journal of Mathematics

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

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

Bradley Duesler and Amanda Knecht

#### 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

**Permanent link to this document**

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

**Keywords**

Rationally connected varieties quasi-algebraically closed

#### 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