2025/08/07 by Garcia-Fritz, Natalia, Pasten, Hector
#11D88 #32Q45 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11D45 #Secondary: 14G05
paper · doi:10.48550/arxiv.2508.05560
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 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.