Xeric varieties
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916885364211712 |
|---|---|
| author | Garcia-Fritz, Natalia Pasten, Hector |
| author_facet | Garcia-Fritz, Natalia Pasten, Hector |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Xeric varieties Garcia-Fritz, Natalia Pasten, Hector Number Theory Algebraic Geometry Complex Variables Primary: 11D45, Secondary: 14G05, 11D88, 32Q45 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. |
| title | Xeric varieties |
| topic | Number Theory Algebraic Geometry Complex Variables Primary: 11D45, Secondary: 14G05, 11D88, 32Q45 |
| url | https://arxiv.org/abs/2508.05560 |