Subfields of ample fields I. Rational maps and definability

TL;DR

This paper investigates the abundance of rational points on smooth curves over ample fields, demonstrating that many points avoid any proper subfield even after rational maps, and explores the structure of definable subsets.

Contribution

It extends Pop's result by showing the existence of many rational points outside any proper subfield after rational maps, and characterizes definable subfields in ample fields.

Findings

Many rational points lie outside any proper subfield after rational maps.

A perfect ample field has no existentially definable proper infinite subfields.

Abstract

Pop proved that a smooth curve C over an ample field K that has a K-rational point has |K| many K-rational points. We strengthen this result by showing that there are |K| many K-rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfect ample field has no existentially definable proper infinite subfields.