相关论文: Transfer principles for Galois cohomology and Serr…

The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension $2$, every torsor over the field for a semisimple,…

We give a criterion for the cohomological dimension of a field, involving norm maps on Milnor K-theory; this criterion was originally formulated by Kato. The theorem we prove is a generalization of a theorem in Serre's book on Galois…

We show that for tame valued fields of equal characteristic with divisible value group, the $C_i$ property lifts from the residue field to the valued field under suitable hypotheses on the residue field. We apply this transfer principle to…

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.…

We remove the assumption "let p be odd or k totally imaginary" from several well-known theorems in Galois cohomology of number fields. For example, we show that the Galois group of the maximal extension of a number field k which is…

Using the recent advancements in the structure of algebraic groups over imperfect fields, we propose a generalization of Serre's Conjecture I and of results that revolve around it. In particular, we prove that the first Galois cohomology…

We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted to 2-dimensional represntations. While the…

Let $K$ be the fraction field of a two-dimensional henselian, excellent, equi-characteristic local domain. We prove a local-global principle for Galois cohomology with finite coefficients over $K$. We use classical machinery from \'etale…

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…

Jacobson developed a counterpart of Galois theory for purely inseparable field extensions in positive characteristic. In his theory, a certain type of derivations replace the role of the generators of Galois groups. This article provides a…

The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We…

K. Kato's conjecture about the cohomological Hasse principle for regular connected schemes $\mathfrak X$ which are flat and proper over the complete discrete valuation rings $\mathcal O_N$ of higher local fields $F_N$ is proven. This…

In 1986, Kato and Kuzumaki stated several conjectures in order to give a diophantine characterization of cohomological dimension of fields. In this article, we first prove a local-global principle in this context for number fields. This…

Assuming that the differential field $(K,\delta)$ is differentially large, in the sense of Le\'on S\'anchez and Tressl, and "bounded" as a field, we prove that for any linear differential algebraic group $G$ over $K$, the differential…

Let K be a complete discretely valued field with residue field k and F be a function field of a curve over K. Let L/F be a Galois extension of degree n. If n is coprime to char(k), then under some assumptions on k(e.g. k is algebraically…

A generalization of Serre's Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover it predicts the set of weights of such…

We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent…

In this paper, we introduce a condition $\mathrm{(F}_m'\mathrm{)}$ on a field $K$, for a positive integer $m$, that generalizes Serre's condition (F) and which still implies the finiteness of the Galois cohomology of finite Galois modules…

The purpose of this paper is to explain how the identities of various fundamental lemmas fall within the scope of the transfer principle, a general result that allows to transfer theorems about identities of p-adic integrals from one…

Ax gave examples of fields of cohomological dimension 1 which are not C_1-fields. Kato and Kuzumaki asked whether a weak form of the C_1-property holds for all fields of cohomological dimension 1 (existence of solutions in extensions of…