Xeric varieties

TL;DR

This paper introduces the concept of xeric varieties, which are smooth projective varieties over number fields with sparse rational points over all finite extensions, exploring their relation to hyperbolicity and rational curves.

Contribution

It systematically studies xeric varieties, linking sparsity of rational points to hyperbolicity and the absence of rational curves, advancing understanding of rational point distribution.

Findings

Xeric varieties have very few rational points over all finite extensions.

The study reveals connections between sparsity, hyperbolicity, and rational curves.

Foundations for further classification of varieties based on rational point distribution.

Abstract

Let $X$ be a smooth projective variety over a number field $k$. The Green--Griffiths--Lang conjecture relates the question of finiteness of rational points in $X$ to the triviality of rational maps from abelian varieties to $X$ and to complex hyperbolicity. Here we investigate the phenomenon of sparsity of rational points in $X$ -- roughly speaking, when there are very few rational points if counted ordered by height. We are interested in the case when sparsity holds over every finite extension of $k$, in which case we say that the variety is \emph{xeric}. We initiate a systematic study of the relation of this property with the non-existence of rational curves in $X$ as well as with certain notion of $p$-adic hyperbolicity.