Drinfeld空间的适度覆盖的de Rham上同调

Cohomologie p-adique de la tour de Drinfeld: le cas de la dimension 1

We compute the p-adic geometric étale cohomology of the coverings of Drinfeld's half-plane, and we show that, if the base field is Q_p, this cohomology encodes the p-adic local Langlands correspondence for 2-dimensional de Rham representations (of weight 0 and 1).

关于双塔中de Rham翻转的性质

On the de Rham flip-flopping in dual towers

本文研究了双塔中rigid分析空间的de Rham和Hyodo-Kato翻转性质，并展示了Drinfeld空间有限级覆盖的上同调作为$\mathbb{GL}_{d+1}(K)$的admissible表示。

Un autre calcul des fonctions inversibles sur l'espace symétrique de Drinfeld

In this article, we give an explicit description of the invertible functions on the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. We identify them with some distribution spaces over the profinite set of $K$-rationnal points of the projective space. The strategy consists of constructing a map from these distributions to the invertible functions following the methods of Schneider-Stuhler, Iovita-Spiess, de Shalit. We show that it is compatible with the isomorphisms they constructed to compute étale and de Rham cohomology in degree $1$ and that this property forces our d…

Cohomologie des courbes analytiques $p$-adiques

Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (…

Comparison of Hyodo-Kato and de Rham Fargues-Fontaine Cohomology Theories

We prove that, for adic étale motives over $\mathbb{C}_p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G_{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sh…

The Geometry of Hida Families I: $Λ$-adic de Rham cohomology

We construct the $Λ$-adic de Rham analogue of Hida's ordinary $Λ$-adic étale cohomology and of Ohta's $Λ$-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of $\mathbf{Q}_p$, we give a purely geometric proof of the expected finiteness, control, and $Λ$-adic duality theorems. Following Ohta, we then prove that our $Λ$-adic module of differentials is canonically isomorphic to the space of ordinary $Λ$-adic cuspforms. In the sequel to this paper, we construct the crystalline counterpart to Hida's ordinary $Λ$-adic étale cohomo…

论p进解析空间的上同调，II：$C_{\ m st}$-猜想

On the cohomology of p-adic analytic spaces, II: The $C_{\rm st}$-conjecture

本文针对光滑dagger簇提出并证明Fontaine猜想的类比，统一了proper与Stein两种极端情形，核心方法是将相关上同调几何化为拟Banach-Colmez空间并利用其精细性质。

Drinfeld半平面覆盖的p-adic局部系统上同调

Cohomologie de systèmes locaux $p$-adiques sur les revêtements du demi-plan de Drinfeld

研究Drinfeld半平面覆盖的p-adic局部系统上同调，将已有2维尖点表示的结果推广至任意权重，发现潜在半稳定非cristabelian表示，并利用Hyodo-Kato上同调与de Rham复形开发proétale上同调计算方法。

The Geometry of Hida Families II: $Λ$-adic $(φ,Γ)$-modules and $Λ$-adic Hodge Theory

We construct the $Λ$-adic crystalline and Dieudonné analogues of Hida's ordinary $Λ$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $Λ$-adic comparison isomorphisms between these cohomologies and the $Λ$-adic de Rham cohomology studied in the prequel to this paper as well as Hida's $Λ$-adic étale cohomology. As applications of our work, we provide a "cohomological" construction of the family of $(φ,Γ)$-modules attached to Hida's ordinary $Λ$-adic étale cohomology by the work of Dee, and we give a new and purely geometric proof of Hida's finitenes and control theorems…

具有循环 $p$-Sylow 子群的覆盖的 de Rham 上同调

The de Rham cohomology of covers with cyclic $p$-Sylow subgroup

本文证明了当有限群 $G$ 具有循环 $p$-Sylow 子群时，光滑射影曲线 $X$ 的 de Rham 上同调的 $G$-结构完全由分歧数据决定，并在两种特定情形下给出了显式公式。

论 $\boldsymbol{\rm B}_{\rm dR}$ 的库默尔 pro-étale 上同调

On the Kummer pro-étale cohomology of $\mathbb B_{\operatorname{dR}}$

本文研究p进域上对数刚性解析簇的p进上同调，计算相关库默尔pro-étale上同调，引入对数$B_{\rm dR}^+$-上同调理论，建立对数德拉姆-étale比较定理并证明两类谱序列的退化性。

p-进微分方程的收敛牛顿多边形 V：局部指标定理

The convergence Newton polygon of a $p$-adic differential equation V : local index theorems

本文研究拟光滑 Berkovich 曲线上带联络的局部自由模，提出绝对局部指标概念，推广 Robba 广义指标与 p-进指数，并给出 de Rham 上同调有限维的精确边界条件及指标公式。

Rigid Cohomology of Drinfeld's Upper Half Space over a Finite Field

In this paper the rigid cohomology of Drinfeld's upper half space over a finite field is computed in two ways. The first method proceeds by computation of the rigid cohomology of the complement of Drinfeld's upper half space in the ambient projective space and then use of the associated long exact sequence for rigid cohomology with proper supports. The second method proceeds by direct computation of rigid cohomology as a direct limit of de Rham cohomologies of a family of strict open neighborhoods of the tube of Drinfeld's upper half space in the ambient rigid-analytic projective space. The re…