Density of rational points on elliptic K3 surfaces

Rational curves on K3 surfaces

“…In characteristic zero, our assertion is shown in [BT,Proposition 2.5]. If char(k) = p > 0, then there exists a possibly ramified extension R of the Witt ring W (k) such that the pair (X, L) lifts to a formal scheme over Spf R by [Del,Corollaire 1.8].…”

Weighted Fano threefold hypersurfaces

“…We prove Theorem 1.1 in Section 4. Finally, we complete the proof of 2 After the early version of this paper, he announced a paper to reinforce his thesis. In his paper, he classified birational transformations into K3 and elliptic fibrations for the cases N = 34, 75, 88, 90 (see [19]).…”

“…where ψ is the natural projection, α 4 is the weighted blow up at the singular point P 4 with weights (1, 1, 5), α 5 is the weighted blow up at the point P 5 with weights (1, 2, 5), β 4 is the weighted blow up with weights (1,1,5) at the point α −1 5 (P 4 ), β 5 is the weighted blow up with weights (1,2,5) at the point α −1 4 (P 5 ), and η is an elliptic fibration. It follows from [6] …”

“…The papers [1], [2], and [8] give us a stimulating result that rational points are potentially dense 3 on smooth Fano 3-folds possibly except double covers of P 3 ramified along smooth sextic surfaces. In the case N = 1, the potential density of rational points on the hypersurface X is proved in [8].…”

K3 Surfaces Over Number Fields with Geometric Picard Number One

“…In this case, we say X has potential density of rational points. In case X C has Picard rank greater than 1, Bogomolov and Tschinkel [2] have shown in many cases that X has potential density of rational points, using the existence of elliptic fibrations on X or large automorphism groups of X. By contrast, we do not know a single example of a K3 surface X/K with geometric Picard number 1 which can be shown to have potential density of rational points; nor is there an example which we can show not to have potential density of rational points.…”