Annette Werner
Inspired by Manin's approach towards a geometric interpretation of Arakelov theory at infinity, we interpret in this paper non-Archimedean local intersection numbers of linear cycles in $P^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to PGL(n).
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We construct a compactification of the Bruhat-Tits building associated to the group PGL(V) by attaching all the Bruhat-Tits buildings of PGL(W) for non-trivial subspaces W of V as a boundary.
In this note we study the special fiber of the Rapoport-Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat-Tits stratification and is intimately related to the Bruhat-Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.
Let $k$ be a complete non-archimedean field (non trivially valued). Given a reductive $k$-group $G$, we prove that hyperspecial subgroups of $G(k)$ (i.e. those arising from reductive models of $G$) are maximal among bounded subgroups. The originality resides in the argument: it is inspired by the case of $\textrm{GL}_n$ and avoids all considerations on the Bruhat-Tits building of $G$.
This work introduces author's theory of Bruhat-Tits buildings over higher dimensional local fields. The theory is illustrated with the buildings for PGL(2) and PGL(3) for one- and two-dimensional local fields.
Consider matrices of order $k+N$ over $p$-adic field determined up to conjugations by elements of $GL$ over $p$-adic integers. We define a product of such conjugacy classes and construct the analog of characteristic functions (transfer functions), they are maps from Bruhat-Tits trees to Bruhat-Tits buildings. We also examine categorical quotient for usual operator colligations.
The Bruhat-Tits stratification for basic unramified $GU(1,n-1)$ Rapoport-Zink spaces at arbitrary parahoric level
本文对签名 $(1,n-1)$ 的基本非分歧酉 Rapoport-Zink 空间在任意抛物子群水平下的约化特殊纤维进行分层，证明了闭分层的光滑性、不可约性并计算了维数，同时利用 Bruhat-Tits 指标描述了分层间的关联关系。
We extend (scheme-theoretic) Bruhat-Tits theory to quasi-reductive groups i.e. with trivial split unipotent radical over discretely valued henselian non-archimedean fields $K$, whose ring of integers is excellent and residue field is perfect, building on Bruhat-Tits I et II and the classification of pseudo-reductive groups by Conrad-Gabber-Prasad. The buildings had been constructed previously by Solleveld in a different manner, but they ultimately coincide with ours. We were motivated by the geometry of affine Grassmannians and give some applications at the end (compare also with previously su…
In a Bruhat-Tits building of split classical type (that is, of type $A_n$, $B_n$, $C_n$, $D_n$, and any combination of them) over a local field, the simplicial volume counts the vertices within the given simplicial distance from a special vertex. This paper aims to study the asymptotic growth of the simplicial volume. A formula of the simplicial volume is deduced from the theory of concave functions. Then the dominant term in its asymptotic growth is found using the theory of $q$-exponential polynomials developed in this paper.
Arithmétique des schémas en groupes de Bruhat-Tits sur un anneau de Dedekind semi-local
本文为半局部戴德金环上布吕阿-蒂茨群概形的上同调研究奠定基础，给出格罗滕迪克-塞尔猜想的简化证明，并推广到半单单连通群情形。
Let $X$ be a smooth projective curve over the complex numbers. We compute the Brauer group of the moduli stack of Bruhat-Tits group scheme $\mathcal{G}$-torsors on $X$. When $g(X) \geq 3$ we compute the Brauer group of the regularly stable locus of the coarse moduli space of semi-stable $\mathcal{G}$-torsors.
Let G be a connected semisimple group over a non-Archimedean local field. For every faithful, geometrically irreducible linear representation of G we define a compactification of the associated Bruhat-Tits building X(G). This yields a finite family of compactifications of X(G) which contains the polyhedral compactification. Moreover, this family can be regarded as a non-Archimedean analogue of Satake compactifications for symmetric spaces.
We construct a compactification of the Bruhat-Tits building associated to the group PGL(V) which can be identified with the space of homothety classes of seminorms on V endowed with the topology of pointwise convergence. Then we define a continuous map from the projective space to this compactification which extends the reduction map from Drinfeld's p-adic symmetric domain to the building.
We describe the stabilizers of points in the Bruhat-Tits building of the group SL with tropical geometry. There are several compactifications of this building associated to algebraic representations of SL. We show that the fans used to compactify apartments in this theory are given by tropical Schur polynomials.
This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry canbe used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.
One of the goals of this article is to describe the isomorphism between Lubin-Tate and Drinfeld towers at the level of their skeletons after taking quotient by $\GL_n (Ø_F)\times Ø_D^\times$ or $I\timesØ_D^\times$ where $Ø_D$ is the maximal order in the division algebra with invariant $\frac{1}{n}$ over $F$ and $I$ a Iwahori subgroup of $\GL_n$. We give applications to the theory of canonical subgroups on Lubin-Tate spaces, the description of spherical Hecke orbits in those spaces, fundamental domains for Hecke correspondences and the Gross-Hopkins period mapping. We also study in details the …
In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…
In this note we make a universal construction of Bruhat-Tits group scheme on wonderful embeddings of adjoint groups in the absolute and relative settings and of adjoint Kac-Moody groups. These have natural classifying properties reflecting the orbit structure on the wonderful embeddings.
Arithmétique des sous-groupes de Bruhat-Tits sur un anneau de valuation discrète hensélien
本文细化了Bruhat-Tits子群的上同调研究，补充1987年工作，并对半单单连通群得到Grothendieck-Serre型结果及拟分裂伴随群情形的障碍精确计算。
The basic locus of unitary splitting Rapoport-Zink spaces with vertex stabilizer level
本文构造了带顶点稳定子水平的酉分裂Rapoport-Zink空间的Bruhat-Tits分层，通过层分裂模型和显式同构研究了其局部与整体结构。
Integral models of moduli spaces of shtukas with deep Bruhat-Tits level structures
本文通过将深 Bruhat-Tits 群概型的整体 shtuka 模空间嵌入所有伴陪群概型模空间的极限，构造了具有良好层映射与 Newton 分层的整模型，并在 Drinfeld 情形恢复已知对象。
Equivariant Chern Classes of Toric Vector Bundles over a DVR and Bruhat--Tits Buildings
本文定义 DVR 上正常环面格式中环面向量丛的等变陈类，借助多面体复形上的分段多项式函数及扩展 Bruhat–Tits 建筑给出组合刻画，并联系非阿基米德 Arakelov 理论。
We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character …
Bruhat-Tits Theory from Berkovich's Point of View. I - Realizations and Compactifications of Buildings
该研究利用Berkovich解析几何研究Bruhat-Tits建筑的紧化，构造了从建筑到对应解析空间的映射并得到紧化族，推广了分裂群情形的结果，还刻画了紧化边界层并证明边界点稳定子群的混合Bruhat分解定理。
Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.
We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functio…
Equivariant vector bundles on complexity-one T-varieties and Bruhat-Tits buildings
本文给出了复杂数为一的正规射影T-簇上环面等变向量丛的组合分类，利用到一般线性群Bruhat-Tits建筑的分段仿射映射推广了已有线丛与环面簇的等变向量丛分类。
Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of valued root groups, thanks to Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p-subgroups thanks to parametrizations of a suitable maximal torus and of corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of…
Bruhat-Tits buildings and $p$-adic period domains
本文研究Bruhat–Tits建筑在Berkovich旗簇中的嵌入与紧化，证明基本局部志村数据下嵌入可经由$p$-进Hodge–Tate周期域分解，并在$\mathrm{GL}_n$互素情形构造连续收缩映射。
We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection …
Given a quasi-reductive group $G$ over a local field $k$, using Berkovich geometry, we exhibit a family of $G(k)$-equivariant compactifications of the Bruhat-Tits building $\mathcal B(G, k)$, constructed and investigated by Solleveld and Lourenço. The compactification procedure consists in mapping the building in the analytification $G^{\mathrm{an}}$ of $G$, then composing this map with the projections from $G^{\mathrm{an}}$ to its (in general non-compact) pseudo-flag varieties $(G/P)^{\mathrm{an}}$, for $P$ ranging among the pseudo-parabolic subgroups of $G$. This generalises previous constru…
Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a Bruhat-Tits group scheme on $X$ which is generically semi-simple and trivial. We show that the étale fundamental group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of torsors under $\mathcal{G}$ is isomorphic to that of the moduli stack $\mathcal{M}_X(G)$ of principal $G$-bundles. For any smooth, noetherian and irreducible stack $\mathcal{X}$, we show that an inclusion of an open substack $\mathcal{X}^\circ$, whose complement has codimension at least two, will induce an isomorphism of étale …
Bruhat-Tits group schemes over higher dimensional base-II
本文证明了在更高维基底上分裂可约的BT群方案是仿射的，并提出了一种更一般的构造方法，结合了扩展的构造技术和结构理论。
The basic locus of ramified unitary Shimura varieties of signature $(n-1,1)$ at maximal vertex level
本文构造了分枝单位ary Rapoport-Zink 空间在签名$(n-1,1)$时的Bruhat--Tits分层，证明其正则性和Cohen--Macaulay性质，并建立与Deligne--Lusztig变种的方案同构。
Wonderful compactifications of Bruhat-Tits buildings
本文研究局部域上分裂半单群对应的欧几里得建筑，证明其最大 Satake-Berkovich 紧化可等变等同于通过嵌入奇妙紧化关联的 Berkovich 空间所得紧化，并探讨无穷远结构关系。
The basic locus of regular ramified unitary Rapoport-Zink spaces at vertex-stabilizer level
本文构造了正则分歧酉 Rapoport-Zink 空间基本轨迹在顶点稳定子层上的 Bruhat-Tits 分层，引入层模型并通过爆破严格变换和线性代数条件两种方式刻画，进而证明各层的光滑性、不可约性并给出显式维数公式。
We define and study a simplicial complex which is a homogeneous space for the group $PGL(2, K)$ over a two-dimensional local field $K$. The complex is a generalization of the tree studied by F. Bruhat, J. Tits, J.-P. Serre and P. Cartier in the 60's and early 70's. Such complex can be canonically attached to the triples $x \in C \subset X$ where $X$ is an algebraic surface, $C$ is an irreducible curve and $x$ is a smooth point on $C$ and $X$. This construction can be used for a description of the isomorphism set of vector bundles on $X$.
Models of hypersurfaces and Bruhat-Tits buildings
本文提出了一种基于布鲁哈-蒂茨建筑的半稳定超曲面模型构造方法，并应用于p进数域上的平面曲线研究。
Wonderful embedding for group schemes in the Bruhat--Tits theory
针对离散赋值亨泽尔域上伴随拟分裂约化群对应的Bruhat--Tits群概形，采用内蕴函子性的Artin--Weil方法变体构造其由大胞腔唯一确定的美妙嵌入，非拟分裂情形通过平展下降实现，搭建了美妙紧化与Bruhat--Tits理论的桥梁。
In our previous paper "Bruhat-Tits theory from Berkovich's point of view. I ? Realizations and compactifications of buildings", we investigated realizations of the Bruhat-Tits building B(G,k) of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry, and we studied in detail the compactifications of the building which arise from this point of view. In this paper, we give a representation theoretic flavor to these compactifications, following Satake's original constructions for Riemannian symmetric spac…
Given an adjoint semisimple group $G$ over a local field $k$, we prove that the maximal Satake-Berkovich compactification of the Bruhat-Tits building of $G$ can be identified with the one obtained by embedding the building into the Berkovich analytification of the wonderful compactification of $G$, extending previous results of Rémy, Thuillier and Werner. In the process, we use the characterisation of the wonderful compactification in terms of Hilbert schemes given by Brion to extend the definition of the wonderful compactification to the case of a non-necessarily split adjoint semisimple grou…
We study the prime-to-$p$ Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat--Tits level at $p$. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considering actions are not transitive. We prove this result by giving negative examples on the question of Bruhat--Colliot-Thélène--Sansuc--Tits or its variant, which is related to the weak approximation on tori over $\mathbb{Q}$.
Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings
本文利用从环面概形扇形到$GL(r)$的分段线性映射，对离散赋值环上的环面等变向量丛进行了分类，并给出了等变分裂判据，为算术几何研究奠定基础。
Relative Bruhat decomposition of wonderful compactification
本文研究了 wonderful group compactification 的 Bruhat 细胞的相对 Bruhat 分解，扩展了 Borel 和 Tits 的经典结果到无穷远处，并针对一般基域进行了探讨。
A quotient of the Bruhat-Tits building classifying maximal matrix orders over curves
本文将布吕阿-蒂茨树的商图构造推广到高维，研究曲线上极大矩阵序的商建筑，明确分裂极大序顶点的结构及与尖点区域的关联，为算术群和序的研究提供几何框架。
On Bruhat-Tits theory over a higher dimensional base
本文推广Bruhat-Tits理论到高维基，通过定义n-有界子群并证明其可模式化，构造了适应正规交叉除数的群概形，并给出了在特征零中的应用。
On tamely ramified $\mathcal G$-bundles on curves
本文证明光滑射影曲线上的仿射 Bruhat-Tits 群概形可表示为驯顺覆盖上约化群概形的不变量，并建立两类扭子及其模叠之间的联系，同时发展等变扭子的局部类型理论。
We study the action on the Bruhat-Tits tree of unit groups of maximal orders in certain quaternion algebras over $\mathbb{F}_q(T)$ and discuss applications to arithmetic geometry and group theory.
A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any…
Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in S, respectively. We construct compactifications of the quotient of X by S-arithmetic subgroups of PGL_d(F). The constructions make delicate use of reductive Borel-Serre spaces for archimedean places and polyhedral and seminorm compactifications at nonarchimedean places. We also briefly discuss a few potential applications of our compacifications…
Normed Fiber Functors
该研究借助Tannakian形式体系，将一般线性群Bruhat-Tits建筑的非阿基米德范数描述推广到非分歧约化群情形，并给出对应仿射群概形的模论刻画。
We define parahoric $\cG$--torsors for certain Bruhat--Tits group scheme $\cG$ on a smooth complex projective curve $X$ when the weights are real, and also define connections on them. We prove that a $\cG$--torsor is given by a homomorphism from $π_1(X\setminus D)$ to a maximal compact subgroup of $G$, where $D\, \subset\, X$ is the parabolic divisor, if and only if the torsor is polystable.
Flatmates and the bounded cohomology of algebraic groups
本文证明非阿基米德局部域上代数群的有界上同调为零，通过解决Bruhat-Tits建筑自同构群的平友猜想，并推导算术群的消失与不变性定理。
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