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arXiv · 2508.05560

Xeric varieties

Natalia Garcia-Fritz·Hector Pasten
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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.

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Natalia Garcia-Fritz, Hector Pasten. 2025-08-07. Xeric varieties. https://arxiv.org/abs/2508.05560

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