A. N. Parshin
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.
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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.
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).
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 $G$.
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。
For F, a finite extension of Q2, and E a quadratic extension of F, I compute the stabilizer in SL(2,F} of a point in the Bruhat-Tits tree of SL(2,E).
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.
We describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to obtain (conjectural) equations for the canonical embedding of Shimura curves.
Two dimensional integral representations via branches of the Bruhat-Tits tree
本文利用Bruhat-Tits树的分支理论研究数域整数环上有限群的二维表示，提出一般计算策略并给出若干特定族的显式公式。
In this paper I study Ramanujan hypergraps and both abstract and explicit constructions is given.
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…
Quantum Unique Ergodicity for Eisenstein Series on Bruhat-Tits Buildings
本文在水平方向上证明了函数域上Eisenstein级数的量子唯一遍历性猜想，通过改进Luo-Sarnak机制并运用谱分解分别处理尖点与Eisenstein贡献。
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。
In this note we study the supersingular locus of the GU(2,2) Shimura variety modulo a prime which is unramified in the imaginary quadratic extension. The supersingular locus of this Shimura variety can be related to the basic Rapoport-Zink space whose special fibre is described by the Bruhat-Tits stratification. The description for this supersingular locus in the case where the prime is inert in imaginary quadratic field is already known to Howard and Pappas by exploiting the exceptional isomorphism. Our method is more direct without using the exceptional isomorphism.
We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sl_n(Q_p). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions.
Formalising the Bruhat-Tits Tree
本文在Lean定理证明器中形式化了布雷哈-蒂茨树，用于验证关于树上调和余链的结论。
The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields
本文研究了函数域上算术群的布鲁hat-蒂茨建筑中不稳定复杂结构，通过修改格雷森的方法，证明了主同余子群的不稳定区域与球形蒂茨建筑的同调等价。
For F an algebraic extension of Q2, the conjugacy classes of invertible, 2-by-2, trace-zero matrices under the action of G := SL2(F) are analyzed relative to the quadratic extension that splits the respective characteristic polynomial. The stabilizer in G of each such matrix is computed as a stabilizer of a simple, Galois invariant path in the Bruhat-Tits Tree of G.
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…
We give an algorithm to explicitly compute the largest subtree, in the local Bruhat-Tits tree for PSL_2(k), whose vertices correspond to orders containing a given suborder H, in terms of a set of generators for H. The shape of this subtree is described, when it is finite, by a set of two invariants. We use our method to provide a full table for the invariants of an order generated by a pair of orthogonal pure quaternions. In a previous work, the first author showed that determining the shape of these local subtrees allows the computation of representation fields, a class field determining the …
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。
For the space $Lat_n$ of all the lattices in a $p$-adic $n$-dimensional linear space we obtain an analog of matrix beta-functions; this beta-function has a degeneration to the Tamagawa zeta-function. We propose an analog of Berezin kernels for $Lat_n$. We obtain conditions of positive definiteness of these kernels and explicit Plancherel formula.
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 …
We consider two sign characters defined on a tamely ramified maximal torus T of a twisted Levi subgroup M of a reductive p-adic group G. We show that their product extends to the stabilizer M(F)_x of any point x in the Bruhat-Tits building of T, and give a formula for this extension. This result is used in the passage between zero and positive depth in the explicit construction of supercuspidal L-packets, as well as in forthcoming work on the Harish-Chandra character formula for supercuspidal representations.
Coefficient systems on the A_2 Bruhat-Tits building
本文探讨了p-adic李群G表示论中的一个猜想，证明了在G的Bruhat-Tits建筑局部区域中，系数系统的链复形精确性可推导出该猜想，尤其在G=SL_3(K)时提供了有力证据。
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。
Let k be a local field and let A be the two-by-two matrix algebra over k. In our previous work we developed a theory that allows the computation of the set of maximal orders in A containing a given suborder. This set is given as a sub-tree of the Bruhat-Tits tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit represen…
$S$-arithmetic groups acting simply transitively on products of Bruhat-Tits trees
针对全定四元数代数上的Bruhat-Tits树，利用Kirschmer和Voight的分类，构造了几乎对所有素理想简单传递作用的同余算术格，并推广到树的乘积，产生新的Ramanujan Cayley图与立方复形。
Let $d \ge 1$. We study a subspace of the space of automorphic forms of $\mathrm{GL}_d$ over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place $\infty$ of $F$, and we consider the subspace $\mathcal{A}_{\mathrm{St}}$ consisting of automorphic forms such that the local component at $\infty$ of the associated automorphic representation is the Steinberg representation (to be made precise in the text). We have two results. One theorem (Theorem 16) describes the constituents of $\mathcal{A}_{\mathrm{St}}$ as automorphic representation an…
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。
Let L be a finite field extension of Q_p and let G be the group of L-rational points of a split connected reductive group over L. We view G as a locally L-analytic group with Lie algebra g. We define a functor from admissible locally analytic G-representations with prescribed infinitesimal character to a category of equivariant sheaves on the Bruhat-Tits building of G. For smooth representations, the corresponding sheaves are closely related to the sheaves constructed by S. Schneider and U. Stuhler. The functor is also compatible, in a certain sense, with the localization of g-modules on the f…
Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-1 map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.
Let $G$ be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic $p$. Let $I$ be a pro-$p$ Iwahori subgroup of $G$ and let $R$ be a commutative quasi-Frobenius ring. If $H=R[I\backslash G/I]$ denotes the pro-$p$ Iwahori-Hecke algebra of $G$ over $R$ we clarify the relation between the category of $H$-modules and the category of $G$-equivariant coefficient systems on the semisimple Bruhat-Tits building of $G$. If $R$ is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler r…
We extend our previous computations for the relative positions of branches of quaternions to the case of local fields of even characteristic. This is a key step to understand the set of maximal orders containing a given suborder, which is useful, for instance, to compute relative spinor images, thus solving the selectivity problem. In our previous work, the results where given in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin-Schreier extensions. It is no longer useful to restrict our attention to orders generated by pure qua…
We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, th…
Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe t…
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 $\mathbb{Q}$.
Covers of Bruhat-Tits trees
本文研究了非阿基米德局部域上元裂覆盖$\operatorname{GL}_2(F)$作用于连通局部有限树的几何结构，并证明了相关对的性质。
Cohomology of the Bruhat-Tits strata in the supersingular locus of the $\mathrm{GU}(1,n-1)$ Shimura variety at a ramified prime
本文计算了分歧素数处 GU(1,n-1) Shimura 簇超奇异轨迹中闭 Bruhat-Tits 层的 ℓ-adic 上同调，证明奇数阶为零、偶数阶为辛群表示，并发现其与交点上同调一致。
Computing quaternionic representations via twisted forms of Bruhat-Tits trees
通过扭曲形式的布鲁哈-蒂茨树研究四元数代数的有限子群表示，计算积分形式集的大小。
Let C be a smooth, projective and geometrically integral curve defined over a finite field F. For each closed point P of C, let R be the ring of functions that are regular outside P, and let K be the completion at P of the function field of C. In order to study groups of the form GL2(R), Serre describes the quotient graph GL2(R)\t, where t is the Bruhat-Tits tree defined from SL2(K). In particular, Serre shows that GL2(R)\t is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property.…
Connecting Kani's Lemma and path-finding in the Bruhat-Tits tree to compute supersingular endomorphism rings
本文提出了一种基于Kani引理和布鲁hat-蒂茨树路径寻找的确定性多项式时间算法，用于计算奇异超椭圆曲线的自同态环。
Let $K$ be an algebraic function field of one variable with constant field $k$ and let $C$ be the Dedekind domain consisting of all those elements of $K$ which are integral outside a fixed place $\infty$ of $K$. When $k$ is finite the group $GL_2(C)$ plays a central role in the theory of Drinfeld modular curves analagous to that played by $SL_2(Z)$ in the classical theory of modular forms. When $k$ is finite (resp. infinite) we refer to a group $GL_2(C)$ as an arithmetic (resp. non-arithmetic) Drinfeld modular group. Associated with $GL_2(C)$ is its Bruhat-Tits tree, $T$. The structure of the …
Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree
本文在Drinfeld半平面上构造非阿基米德Poincaré级数，证明其收敛性与非零性，并用于生成模形式空间，推广了Kurihara的方法。
Let K be a function field with constant field k and let "infinity" be a fixed place of K. Let C be the Dedekind domain consisting of all those elements of K which are integral outside "infinity". The group G=GL_2(C) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat-Tits tree, T. Classical Bass-Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) G\T. The shape of this quot…
Eisenstein scattering and Plancherel decomposition on cuspidal Bruhat-Tits quotients
本文针对Bruhat-Tits树的算术商，建立了Eisenstein变换与Jacobi算子散射变换的显式酉对应，推导出矩阵值Maass-Selberg公式，并利用共振计算区分特征值与散射极点。
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 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。
In this paper, we study the cohomology of the ramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ by using the Bruhat-Tits stratification on its special fiber. As such, we apply the same method that we developped for the unramified case in two previous papers. More precisely, we first investigate the cohomology of a given closed Bruhat-Tits stratum. It is isomorphic to a generalized Deligne-Lusztig variety which is in general not smooth, and is associated to a finite group of symplectic similitudes. We determine the weights of the Frobenius and most of the unipotent representations…
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.
The prime geodesic theorem for cycles in Bruhat-Tits buildings is applied to unit groups of division algebras to derive new asymptotic assertion on class numbers of orders in imaginary quadratic fields.
Let G be a semisimple quasi-split group defined over a global function field. We express the relative Tamagawa number of G in terms of local data including the number of types of special vertices in one orbit of the Bruhat-Tits building over some completion with respect to a place and the class number of G at that place.
Diophantine Approximation in local function fields via Bruhat-Tit trees
基于 Bruhat-Tits 树的算术商理论，获得了全局函数域上丢番图逼近的 Dirichlet 风格定理，并计算了相关常数与逼近集合测度。
We compute the Euler-Poincaré characteristic of quotients of the Bruhat-Tits building of PGL(n) under the action of arithmetic groups arising from central division algebras over rational function fields of positive characteristic. We use this result to determine the structure of the quotient simplicial complex in certain cases.
We determine the maximal eigenvalue of the p-adic curvature transformations on Bruhat-Tits buildings, and we give an essentially optimal upper bound on the minimal non-zero eigenvalue of these transformations.
The exceptional group $G_2$, when constructed over $Q_p$, may at the same time be considered as the group of automorphisms of an octonion algebra over $Q_p$, or alternatively it may be constructed as a Chevalley group from the root diagram of the appropriate Lie algebra, $g_2$. Each construction has its benefits and drawbacks, and the first part of this work focuses on drawing parallels between the two and describing certain group structures in terms of both. In particular, we explicitly identify the Chevalley generators of $G_2$ as automorphisms of octonions. In the second part of this work w…
The analogue of the Bruhat-Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.
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