The Chow group of a Ch{â}telet surface over a number field

关于数字段上椭圆面的Chow群

On the Chow group of elliptic surfaces over number fields

本文研究数字段上椭圆面的Chow群，证明其核是一个有限生成群。

Higher Chow Cycles on an Abelian Surface and a non-Archimedean analogue of the Hodge-D-conjecture

We construct new indecomposable elements in the higher Chow group CH2(A,1) of a principally polarized Abelian surface over a non Archimedean local field, which generalize an element constructed by Collino. These elements are constructed using a generalization, due to Birkenhake and Wilhelm, of a classical construction of Humbert, along with some recent work of Bogomolov, Hassett and Tschinkel on deformations of rational curves on a K3 surface. They can be used to prove the non-Archimedean Hodge-D-conjecture - namely, the surjectivity of the boundary map in the localization sequence - in the ca…

Relations among Heegner Cycles on families of abelian surfaces

We compute relations of rational equivalence among special codimensional two cycles on families of abelian surfaces using elements of higher chow groups.

在$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖上构造高 Chow 环，第二部分

Construction of higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$, Part II

本文研究了在$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖上构造高 Chow 环，并证明其生成不可分解部分的子群。

循环覆盖上的高阶Chow环的构造$\mathbb{P}^1 \times \mathbb{P}^1$

Construction of Higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$

本研究针对$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖曲面族，构造了$(2, 1)$型高阶Chow环，通过计算超越调节映射的像，证明其在$\mathbb{Z}$上线性无关且生成$\operatorname{rank} \ge 36 \cdot φ(N)$的子群。

Selmer group associated to the Chow group of certain codimension two cycles

Let $X$ be a surface with geometric genus and irregularity zero which is defined over a number field $K$. Let $\mathscr{X}$ denote a smooth spread of $X$ over the spectrum of a Zariski open subset in the spectrum of the ring of integers and $A^2$ stands for the group of algebraically trivial cycles on schemes modulo rational equivalence. If $j^*: A^2(\mathscr{X})\to A^2(X)$ be the flat pull-back corresponding to the embedding $j:X\hookrightarrow \mathscr{X}$ then we prove that $\im(j^*)(K)/A^2(\mathscr{X})(K)$ is a torsion group. Here $\im(j^*)(K)$, $A^2(\mathscr{X})(K)$ stand for the cycles f…

Relative K-theory and class field theory for arithmetic surfaces

In this paper we extend the unramified class field theory for arithmetic surfaces of K. Kato and S. Saito to the relative case. Let X be a regular proper arithmetic surface and let Y be the support of divisor on X. Let CH_0(X,Y) denote the relative Chow group of zero cycles and let \tilde π_1^t(X,Y)^ {ab} denote the abelianized modified tame fundamental group of (X,Y) (which classifies finite etale abelian covings of X-Y which are tamely ramified along Y and in which every real point splits completely). THEOREM: There exists a natural reciprocity isomorphism rec: CH_0(X,Y) --> \tilde π_1^t(X,Y…

On the cycle class map for zero-cycles over local fields

We study the Chow group of zero-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic different from l. The same statement holds for semistable K3 surfaces defined over C((t)), but does not hold in general for surfaces over strictly local fields.

K3曲面上的超椭圆曲线

Hyperelliptic curves on $K3$ surfaces

该研究证明皮卡秩为1的K3曲面上任意多个一般点可由超椭圆曲线连接，得到小对角的Beauville–Voisin分解及挠阶新界，还将Corvaja–Zucconi定理推广到任意次数K3曲面，证明其二次点在有限扩张后稠密。