Some aspects of rational points and rational curves

TL;DR

This paper surveys recent advances in methods for constructing rational points and rational curves on algebraic varieties, focusing on descent, fibration, and their applications over different fields.

Contribution

It provides a comparative overview of descent and fibration methods in both number theory and algebraic geometry contexts.

Findings

Advances in constructing rational points over number fields.

Progress in understanding rational curves on real varieties.

Discussion on rational points over function fields of p-adic curves.

Abstract

Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context (rational points over number fields) and in an algebro-geometric one (rational curves on real varieties), and discuss the question of rational points over function fields of p-adic curves.