Canonical vector heights on K3 surfaces with Picard number three -- addendum

Neron-Severi group preserving lifting of K3 surfaces and applications

For a K3 surface of finite height over a field of odd characteristic, there exists a smooth lifting to the ring of Witt vectors such that the reduction map from the Picard group of the generic fiber to the Picard group of the special fiber is isomorphic. In this paper, using this result, we give a criterion for a K3 surface of finite height over a field of odd characteristic to be an Enriques K3 surface or to be a K3 surface in terms of the Neron-Severi lattice. Then we show every Kummer surface has an Enriques involution. We also give a classification of K3 surfaces of Picard rank of 20 over …

双椭圆Picard曲线的Ceresa圈

Ceresa cycles of bielliptic Picard curves

本文证明了双椭圆Picard曲线的Ceresa圈为挠群当且仅当对应椭圆曲线上的点为挠点，揭示了其高度的Northcott性质，并利用Chow motive同构完成了证明。

Tamagawa numbers of diagonal cubic surfaces of higher rank

We consider diagonal cubic surfaces defined by an equation of the form ax^3+by^3+cz^3+dt^3 = 0. Numerically, one can find all rational points of height < B for B in the range of up to 100 000, thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previo…

Existence of supersingular reduction for families of K3 surfaces with large Picard number in positive characteristic

We study non-isotrivial families of $K3$ surfaces in positive characteristic $p$ whose geometric generic fibers satisfy $ρ\geq21-2h$ and $h\geq3$, where $ρ$ is the Picard number and $h$ is the height of the formal Brauer group. We show that, under a mild assumption on the characteristic of the base field, they have potential supersingular reduction. Our methods rely on Maulik's results on moduli spaces of $K3$ surfaces and the construction of sections of powers of Hodge bundles due to van der Geer and Katsura. For large $p$ and each $2\leq{h}\leq10$, using deformation theory and Taelman's meth…

Rational points of bounded height on general conic bundle surfaces

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

Hecke特征标与Picard曲面的Eisenstein上同调的Bloch-Beilinson猜想

Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces

本文研究Hecke特征标与Picard曲面的Eisenstein上同调之间的Bloch-Beilinson猜想，通过构造Hodge结构延伸并计算双延伸高度来验证其非平凡性。