Endoscopy and cohomology of a quasi-split U(4)

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

We continue our study of the poles of local Langlands L-functions through the theory of induced from supercuspidal representations of quasi-split groups. Here we study the odd special orthogonal groups, and hence determine poles of Rankin product L-functions. The pole of the intertwining operator is determined in terms of the theory of orbital integrals. This gives a description of the poles in terms of twisted endoscopy, as in previous cases. We use the language of functorial transfer to give precise descrption of the pole in terms of the local components of the global transfer, which has now…

非齐次Tate上链的杯积，以及分裂于循环扩张的局部域上环面的Galois上同调

Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensions

本文给出Tate上同调中杯积的非齐次上链公式，并用于计算分裂于循环扩张的局部域上环面的一阶上同调类的显式上循环。

On certain cohomology groups attached to $\mathfrak{p}^{\infty}$-towers of quaternionic Hilbert modular varieties

For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $π$ of $D^*(\mathbb{A})$ this yields a natural construction…

论带挠系数的希尔伯特模簇的平展上同调

On the étale cohomology of Hilbert modular varieties with torsion coefficients

本文研究带挠系数的希尔伯特模簇平展上同调，借鉴酉志村簇方法得到一般情形下的中间次数集中零化定理与非一般情形的次数界，借助几何Jacquet--Langlands函子性证明并应用于p进局部朗兰兹对应。

Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the étale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the cont…

The continuous cohomology of period domains over local fields

In this paper we consider period domains over local fields for quasi-split reductive groups. We compute the continuous l-adic cohomology with compact support of them in the case of a basic isocrystal. This paper is a continuation of [O2] where we considered the etale cohomology with torsion coefficients of these spaces.

$l$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction…

关于具有非准分裂局部群的简单Shimura簇的上同调

On the cohomology of simple Shimura varieties with non quasi-split local groups

本文研究了非准分裂局部群下简单Shimura簇坏约化的Scholze测试函数，证明了其扭曲轨道积分的消没性质及伪稳定基变换的存在性，并推导出在稳定迹公式和局部Hasse-Weil zeta函数中的应用。

全分裂情形下Hilbert模簇的高阶Hida理论

Higher Hida theory for Hilbert modular varieties in the totally split case

研究全实域F中素数p完全分裂时，Hilbert模簇X上自守层的凝聚上同调的p进性质，构造了p进插值普通部分的模M^q。

Galois cohomology and profinitely solitary Chevalley groups

For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. Assuming Grothendieck rigidity, we essentially solve the problem by Galois cohomological means.

关于具有给定二次代数与上同调的数域扩张

On extensions of number fields with given quadratic algebras and cohomology

本文提出有限表示 pro-p 群的表示判据以计算上同调并构造高维 mild 群商，并将其刻画为带指定分歧与分裂的 p-有理域 Galois 群。

Four-fold Massey products in Galois cohomology

In this paper, we develop a new necessary and sufficient condition for the vanishing of 4-Massey products of elements in the mod-2 Galois cohomology of a field. This new description allows us to define a splitting variety for 4-Massey products, which is shown in the Appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such 4-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.

p进Chevalley群的上同调

Cohomology of $p$-adic Chevalley groups

本文在Coxeter数标准假设下，利用分次李代数的新展示和正特征旗簇凝聚上同调，计算了p进Chevalley群及其Iwahori子群的上同调代数，并讨论了与Morava稳定化子群相关的GL_n(K)内形式。

On the cohomology of integral $p$-adic unipotent radicals

Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.