On the Cohomology of Locally Symmetric Spaces and of their Compactifications

A resolution of singularities for Drinfeld's compactification by stable maps

Drinfeld's relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting of stable maps from nodal deformations of the curve into twisted flag varieties. As an application, we prove that the twisted intersection cohomology sheaf on Drinfeld's compactification is universally locally acyclic over the moduli stack of $G$-bundles at points sufficiently antidominant relative to their defect.

Analysis on Symmetric and Locally Symmetric Spaces (Multiplicities, Cohomology and Zeta functions)

The goal of the course was a review of results mainly due to M. Olbrich and the first author. We consider a discrete cocompact subgroup $Γ$ of a semisimple Lie group $G$. We relate the group cohomology of $Γ$ with coefficients in the maximal globalization of a representation of $G$ with the multiplicities of unitary representations of $G$ in $L^2(G/Γ)$. Explicit calculations are given in the case $G=SL(2,R)$. In the rank-one case we state a version of the Selberg trace formula, discuss the singularities of the Selberg zeta-functions, and state their relation with the cohomology of $Γ$ in princ…

非阿贝尔局部紧群的Bohr紧化与Chu对偶

Bohr compactification and Chu duality of non-abelian locally compact groups

本研究聚焦非阿贝尔局部紧群的Bohr紧化，刻画其Bohr紧化与Chu对偶或酉拟对偶拓扑同构的条件，修正了文献中部分错误结论。

L-modules and the Conjecture of Rapoport and Goresky-MacPherson

Consider the middle perversity intersection cohomology groups of various compactifications of a Hermitian locally symmetric space. Rapoport and independently Goresky and MacPherson have conjectured that these groups coincide for the reductive Borel-Serre compactification and the Baily-Borel-Satake compactification. This paper describes the theory of L-modules and how it is used to solve the conjecture. More generally we consider a Satake compactification for which all real boundary components are equal-rank. Details will be given elsewhere (math.RT/0112251). As another application of L-modules…

Sur la cohomologie des compactifications de variétés de Deligne-Lusztig

In this article, we study the étale cohomology of the compactification of Deligne-Lusztig varieties associated to a Coxeter element. We prove a result for the integral coefficients in the case of general linear group $GL_d$, and we conjecture that the similar result holds for general reductive groups.

Continuous group cohomology with coefficients in locally analytic vectors of admissible $ \mathbb{Q}_p $-Banach space representations

We show that the continuous cohomology groups of a $ p $-adic reductive group with coefficients in the locally analytic vectors of an admissible $ \mathbb{Q}_p $-Banach space representation are homeomorphic to those with coefficients in the Banach space representation itself. Moreover, we deduce that the canonical topologies on those continuous cohomology groups are Hausdorff and are the uniquely determined finest locally convex topologies.

Boundary behavior of special cohomology classes arising from the Weil representation

In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p,q)$. This correspondence is realized using theta functions associated to explicitly constructed "special" Schwartz forms. Furthermore, the theta functions give rise to generating series of certain "special cycles" in $X$ with coefficients. In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the t…

局部平面整曲线的紧化雅可比的上同调

Cohomology of compactified Jacobians for locally planar integral curves

本文研究局部平面整曲线紧化雅可比的上同调，探讨Ngô支持定理、perverse滤波及与希尔伯特方案的联系。

${\rm GL}_n$ 的 Deligne--Lusztig 簇的上同调与几何

Cohomology and geometry of Deligne--Lusztig varieties for ${\rm GL}_n$

本文描述了 ${\rm GL}_n$ 的 Deligne--Lusztig 簇的光滑紧化上结构层的上同调，进而得到其模 $p^m$ 与整 $p$-进 étale 上同调，并证明 Zariski 闭包具有伪有理奇点。

On locally analytic vectors of the completed cohomology of modular curves

We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight $k$, we show its corresponding Galois representation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.

Automorphic Lefschetz properties via $L^2$ cohomology

In this paper one proves a special case of a conjecture by Nicolas Bergeron. This conjecture is a kind of automorphic Lefschetz property. It relates the primitive cohomology of a locally symmetric manifolds modeled on $U(p,q+r)$ to the primitive cohomology of some of its totally geodesic submanifolds that are locally symmetric and modeled on $U(p,q)$.

四元数Shimura曲线完成余弦学中局部解析向量的研究

Locally analytic vectors in the completed cohomology of quaternionic Shimura curves

本文研究四元数Shimura曲线完成余弦学中局部解析向量的性质及其与Hecke特征值的关系，并探讨其在晶体质地情况下的Hodge滤波检测问题。

On torsion in the cohomology of locally symmetric varieties

The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\GL_n$ over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\GL_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of …

Solid locally analytic representations of $p$-adic Lie groups

We develop the theory of locally analytic representations of compact $p$-adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard's isomorphisms between continuous, locally analytic and Lie algebra cohomology to solid representations. We also prove a comparison result between the group cohomology of a solid representation and of its analytic vectors.

Complexes d'intersection des compactifications de Baily-Borel : Le cas des variétés de Siegel

In this work, we calculate the trace of a Hecke correspondance composed with a power of the Frobenius endomorphism on the fibre of the intersection complexes of the Baily-Borel compactification of a Siegel modular variety. Our main tool is Pink's theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncation of the full direct image. More generally, we are able to define analogs in positive c…