Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters

Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map

We construct a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field and prove a comparison with the rigid syntomic regulator. For smooth projective schemes we further relate the relative Chern character to the etale p-adic regulator via the Bloch-Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers and generalizes it to all smooth projective schemes as above.

Kato conjecture and motivic cohomology over finite fields

For an arithmetical scheme X, K. Kato introduced a certain complex of Gersten-Bloch-Ogus type whose component in degree a involves Galois cohomology groups of the residue fields of all the points of X of dimension a. He stated a conjecture on its homology generalizing the fundamental exact sequences for Brauer groups of global fields. We prove the conjecture over a finite field assuming resolution of singularities. Thanks to a recently established result on resolution of singularities for embedded surfaces, it implies the unconditional vanishing of the homology up to degree 4 for X projective …

2-部分的Bloch-Kato猜想及某些椭圆曲线的K₂的不可除结果

The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves

本文研究特定整数u的2-部分Bloch-Kato猜想，分析Selmer群的2- torsion，并构造K₂的不可除元素，通过Merkur'ev-Suslin同构证明其不可除性，同时探讨2-可除性界及数值验证。

Swan Conductors and Torsion in the Logarithmic de Rham Complex

Using a recent work of K.Kato and T.Saito we give a logarithmic version of a formula of S.Bloch in ramification theory.

Tamagawa numbers of some crystalline representations

The purpose of this article is to give a proof of the $C_{EP,F}(V)$ conjecture for some semi-stable representations and of the $δ_{\Zp}(V)$ conjecture for some crystalline representations. There are two major ingredients: first, the computation of Tamagawa numbers using a variation on results of Fontaine-Laffaille and of Bloch-Kato, and second some continuity results for Bloch-Kato's exponential which were proved by Perrin-Riou.

On motivic cohomology with Z/l coefficients

In this paper we give a proof of the Bloch-Kato conjecture relating motivic cohomology and etale cohomology. It is a corrected version of the paper with the same title which posted earlier.

Breuil-Kisin 函子的正合性质与 Bloch-Kato Selmer 群

Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups

本文证明Breuil-Kisin函子在半稳定表示格上的正合性，并利用该结果通过强可分模范畴中的Ext群计算积分Bloch-Kato Selmer群，同时定义了与Galois表示函子交换的张量积，应用于阿贝尔簇的杯积映射分解。

Multiplicative properties of the multiplicative group

We give a few properties equivalent to the Bloch-Kato conjecture (now the norm residue isomorphism theorem).

Hasse principles for higher-dimensional fields

For schemes X over global or local fields, or over their rings of integers, K. Kato stated several conjectures on certain complexes of Gersten-Bloch-Ogus type, generalizing the fundamental exact sequence of Brauer groups for a global field. He proved these conjectures for low dimensions. We prove Kato's conjecture over number fields. In particular this gives a Hasse principle for function fields F over a number field K, involving the corresponding function fields over all completions of K. We get the same results over global fields K of positive characteristic, for coefficients invertible in K…

Beilinson's Hodge conjecture for smooth varieties

Consider the cycle class map cl_{r,m} : CH^r(U,m;\Q) \to ΓH^{2r-m}(U,\Q(r)), where CH^r(U,m;\Q) is Bloch's higher Chow group (tensored with \Q) of a smooth complex quasi-projective variety U, and H^{2r-m}(U,\Q(r)) is singular cohomology. We study the image of cl_{r,m} in terms of kernels of Abel-Jacobi maps. When r=m, we deduce from the Bloch-Kato theorem that the cokernel of cl_{r,m} at the generic point of U is the same for integral or rational coefficients.

通过Iwasawa轨迹计算CM曲线的幂单Selmer维数

Lie weights for CM curves and unipotent Selmer dimensions

该研究针对几何CM雅可比的亏格≥2的Q上光滑射影曲线，证明了Bloch-Kato Selmer概形的维数不等式，回答了Kim猜想的特殊情形，核心方法基于Iwasawa理论与特征重数界估。

关于Rankin–Selberg动机的Selmer群的有界性综述

Survey on bounding Selmer groups for Rankin--Selberg motives

本文综述作者此前关于Rankin–Selberg动机的Beilinson–Bloch–Kato猜想研究，并结合X. Zhu的范畴局部Langlands纲领提出新观察与推测。

Comparison of Dualizing Complexes

We prove that there is a map from Bloch's cycle complex to Kato's complex of Milnor K-theory, which induces a quasi-isomorphism from étale sheafified cycle complex to the Gersten complex of logarithmic de Rham--Witt sheaves. Next we show that the truncation of Bloch's cycle complex at -3 is quasi-isomorphic to Spiess' dualizing complex.

Koszul对偶与Galois上同调

Koszul duality and Galois cohomology

证明了模剩余同态$K^M(F)/l \to H^*(G_F,Z/l)$的Bloch-Kato猜想是在模$l$的Milnor K-理论代数$K^M(F)/l$是Koszul的假设下，从其（部分已知的）低次部分推出的.这个结论是幂零（余）代数上同调和Koszulity的一般结果的一个例子。

Invitation to higher local fields, Part I, section 4: Cohomological symbol for henselian discrete valuation fields of mixed characteristic

This is a sketch of main steps of the proof of Bloch--Kato's theorem which states that the norm residue homomorphism K_q(K)/m\to H^q(K,\Bbb Z/m(q)) is an isomorphism for a henselian discrete valuation field K of characteristic 0 with residue field of positive characteristic.