伪刚性空间上$(φ,Γ)$-模的上同调

Galois Cohomology for Lubin-Tate $(φ_q,Γ_{LT})$-modules over Coefficient rings

The classification of the local Galois representations using $(φ,Γ)$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $(φ_q,Γ_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $(ψ_q,Γ_{LT})$-modules. We also comput…

Lubin-Tate $(\varphi_L,\Gamma_L)$-模的解析上同调的有限性

Finiteness of analytic cohomology of Lubin-Tate $(φ_L,Γ_L)$-modules

本文证明了由affinoid代数参数化的$L$-解析$(\varphi_L,\Gamma_L)$-模族的解析上同调具有有限性和基变换性质，并通过广义Herr复形给出显式描述。

关于循环群上有限型G-模的H¹上同调群

On cohomology groups $ H^{1} $ of $ G-$modules of finite type over cyclic groups

本文研究循环群上有限生成G-模的H¹上同调群与Herbrand商的关系，应用于二次数域单位群、Pell方程和数域类数。

Cohomology and Duality for (phi,Gamma)-modules over the Robba ring

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated etale (phi,Gamma)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily etale (phi,Gamma)-modules over the Robba ring, of the Euler-Poincare characteristic formula and Tate local duality for p-adic representations. These results are expected to intervene in the duality theory for Selmer groups associated to de Rham representations.

A comparison between compactly supported rigid and $\pmb{\mathscr{D}}$-module cohomology

The goal of this article is to prove a comparison theorem between rigid cohomology and cohomology computed using the theory of arithmetic $\mathscr{D}$-modules. To do this, we construct a specialisation functor from Le Stum's category of constructible isocrystals to the derived category of arithmetic $\mathscr{D}$-modules. For objects `of Frobenius type', we show that the essential image of this functor consists of overholonomic $\mathscr{D}^\dagger$-modules, and lies inside the heart of the dual constructible t-structure. We use this to give a more global construction of Caro's specialisation…

Cohomology of arithmetic families of (phi,Gamma)-modules

We prove the finiteness and compatibility with base change of the (phi,Gamma)-cohomology and the Iwasawa cohomology of arithmetic families of (phi,Gamma)-modules. Using this finiteness theorem, we show that a family of Galois representations that is densely pointwise refined in the sense of Mazur is actually trianguline as a family over a large subspace. In the case of the Coleman-Mazur eigencurve, we determine the behavior at all points.

On the faithfulness of parabolic cohomology as a Hecke module over a finite field

In this article we prove conditions under which a certain parabolic group cohomology space over a finite field F is a faithful module for the Hecke algebra of Katz modular forms over an algebraic closure of F. These results can e.g. be used to compute Katz modular forms of weight one with methods of linear algebra over F. This is essentially Chapter 3 of my thesis.

Invitation to higher local fields, Part II, section 6: Phi-Gamma-modules and Galois cohomology

This is is a survey of applications of Fontaine's theory of p-adic representations of local fields (Phi-Gamma-modules) to Galois cohomology of local fields and explicit formulas for the Hilbert symbol in relation with two-dimensional local objects.

Breuil-Kisin Modules via crystalline cohomology

For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic étale cohomology $H^i_{\mathrm{ét}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.

高余维数中的同余模与伽罗瓦上同调中的 zeta 线

Congruence modules in higher codimension and zeta lines in Galois cohomology

本文基于高余维数同余模概念，提出了检测局部环正则性的准则与形变下同余模变化的精细结果，并应用于由 Hilbert 模形式伴随动机导出的 Galois 上同调群中构造典范线。

Quasi-abelian crossed modules and nonabelian cohomology

We extend the work of M.Borovoi on the nonabelian Galois cohomology of linear reductive algebraic groups over number fields to a general base scheme. As an application, we obtain new results on the arithmetic of such groups over global function fields.