Nicole Kraemer
In this article, we study local models associated to certain Shimura varieties. In particular, we present a resoultion of their singularities. As a consequence, we are able to determine the alternating semisimple trace of the geometric Frobenius on the sheaf of nearby cycles.
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Special cycles on the basic locus of unitary Shimura varieties at ramified primes
本文研究分歧素数处酉志村簇基本轨迹上的特殊闭链及其 Rapoport-Zink 空间局部类比，分析其支集并计算维数。
In this paper, we study special cycles on the Krämer model of $\mathrm{U}(1,1)(F/F_0)$-Rapoport-Zink spaces where $F/F_0$ is a ramified quadratic extension of $p$-adic number fields with the assumption that the $2$-dimensional hermitian space of special quasi-homomorphisms is anisotropic. We write down the decomposition of these special cycles and prove a version of Kudla-Rapoport conjecture in this case. We then apply the local results to compute the intersection numbers of special cycles on unitary Shimura curves and relate these intersection numbers to Fourier coefficients of central deriva…
We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good" $p$-adic integral models of these Shimura varieties and study their 'etale local structure. In this paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties in our previous paper math.AG/0607130.
Regular models of ramified unitary Shimura varieties at maximal parahoric level
本文利用分裂模型的思想，定义并研究了在分歧素数处具有极大抛物水平结构的符号为$(n-1,1)$的酉Shimura簇的半稳定模型，证明了该情形下的真正分裂模型是平坦且具有半稳定约化的。
On the moduli description of ramified unitary local models of signature $(n-1,1)$
本文在特征非2条件下显式刻画签名 $(n-1,1)$ 分歧酉局部模型的特殊纤维，证实 Smithling 猜想，并推广至 Pappas–Zhu 模型与 Shimura 簇整模型。
Pappas's conjecture on ramified unitary local models
本文证明了分歧酉群楔形局部模型在特定素数条件下的平坦性，确认了Pappas猜想，方法是将问题归约到幂零轨道闭包和Kempf分解，并利用Koszul复形简化计算。
Spherical representations of unitary groups at ramified places and the arithmetic inner product formula
本文研究了局部域上偶单位群的可允许表示，其中二次扩张是分歧的，并考虑了在unimodular晶格稳定子作用下不变的向量以及相应单位Shimura品种积分模型的一些性质。作为直接应用，本文改进了算术内积公式，允许局部根数为(-1)的处为分歧。
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。
Semi-stable models for some unitary Shimura varieties over ramified primes
本文针对签名 $(2,n-2)$ 的酉群相关的 Shimura 簇，通过显式分解局部模型，在虚二次域分歧的奇素数 $p$ 上构建了正则 $p$-adic 整模型。
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。
Semi-stable and splitting models for unitary Shimura varieties over ramified places. I
本文针对偶维酉群对应的志村簇，在奇素数分歧情形下构造了 $s=1$ 的光滑及 $s=2,3$ 的正则 $p$-进整模型，方法是对局部模型作显式消解，并给出模空间解释。
Wildly ramified unitary local models for special parahorics. The odd dimensional case
本文为奇数维野分歧酉相似群构造了特殊 parahoric 级局部模型，利用 Bruhat-Tits 理论分析了两类共轭子群，证明了其平滑性或正规且 Cohen-Macaulay 性质，并建立了与 Scholze-Weinstein v-层模型的联系及显式模空间解释。
Local models are schemes which are intended to model the étale-local structure of p-adic integral models of Shimura varieties. Pappas and Zhu have recently given a general group-theoretic construction of flat local models with parahoric level structure for any tamely ramified group, but it remains an interesting problem to characterize the local models, when possible, in terms of an explicit moduli problem. In the setting of local models for ramified, quasi-split GU_n, work towards an explicit moduli description was initiated in the general framework of Rapoport and Zink's book and was subsequ…
Local models are schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at Q_p is ramified, quasi-split GU_n, Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper we proved that their new local models are topologically flat when n is o…
In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in previous work, following the stabilization…
Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Q_p are ramified, quasi-split GU_n, Pappas and Rapoport have added new conditions, the so…
Unitary Shimura varieties at ramified primes and arithmetic transfer
本文研究分歧素数处酉 Shimura 簇，构造局部模型比较同构，提升至 p-可除群与 Rapoport-Zink 空间，并完全证明算术转移猜想。
Semi-stable and splitting models for unitary Shimura varieties over ramified places. II
本文针对符号为 $(n-1,1)$ 的酉群志村簇，在分歧奇素数 $p$ 上构造具有简单模空间描述的 $p$-进整分裂模型，并证明其正规平坦且为 Cohen-Macaulay，进一步通过一次 blow-up 得到半稳定模型。
Arithmetic level raising for certain quaternionic unitary Shimura variety
本文证明了在分支情况下四维辛群的算术提升定理，为贝利森-布洛赫-卡托猜想中的某些兰金-塞尔伯格动机提供了关键步骤，同时可视为特殊纤维 Shimura 变种在分支特征下的伊哈拉引理或塔特猜想类比。
Local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain Shimura varieties, with parahoric level structure. We investigate the local models for groups of the form Res_{F/\Q_p} GL_n and Res_{F/\Q_p} GSp_2g where F/Q_p is a totally ramified extension, as defined by Pappas and Rapoport, and show that they are topologically flat. In the linear case, flatness can be deduced from this.
More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case
本文针对 p 进域二次分歧扩张上的酉群，构造了带仿射仿射层次的正则形式模空间及其间的对应，提出算术转移猜想，并在最低维情形下给出证明。
Wildly Ramified Rigid $G_2$-Local Systems
本文针对正特征下斜率分子为1的野分歧刚性$G_2$-局部系统，提供了分类所需的关键结果，方法与早期不规则连接分类类似但部分不变量获取方式不同。
This paper is a continuation of our paper math.AG/0006222. We study the reduction of certain PEL Shimura varieties with parahoric level structure at primes p at which the group that defines the Shimura variety ramifies. We describe "good" $p$-adic integral models of these Shimura varieties and study their 'etale local structure. In particular, we exhibit a stratification of their (singular) special fibers and give a partial calculation of the sheaf of nearby cycles.
We analyze the geometry of the supersingular locus of the reduction modulo p of a Shimura variety associated to a unitary similitude group GU(1,n-1) over Q, in the case that p is ramified. We define a stratification of this locus and show that its incidence complex is closely related to a certain Bruhat-Tits simplicial complex. Each stratum is isomorphic to a Deligne-Lusztig variety associated to some symplectic group over F_p and some Coxeter element. The closure of each stratum is a normal projective variety with at most isolated singularities. The results are analogous to those of Vollaard/…
The aim of these notes is to generalize Laumon's construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article "On the geometric Langlands conjecture" by Frenkel, Gaitsgory and Vilonen [10] carry over to our situation. We show that our sheaves descend to the moduli space of parabolic …
Local models are schemes defined in linear algebra terms that describe the 'etale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study in detail certain modifications of the local models for G=R_{E/F}GL(n), with E/F a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We show that the new lo…
The obstruction to the local-global principle for a hermitian lattice (L, H) can be quantified by computing the mass of (L, H). The mass formula expresses the mass of (L, H) as a product of local factors, called the local densities of (L, H). The local density formula is known except in the case of a ramified hermitian lattice of residue characteristic 2. Let F be a finite unramified field extension of Q_2. Ramified quadratic extensions E/F fall into two cases that we call Case 1 and Case 2. In this paper, we obtain the local density formula for a ramified hermitian lattice in Case 1, by const…
A proof of the Kudla-Rapoport conjecture for Krämer models
本文证明了分歧处酉 Rapoport-Zink 空间 Krämer 模型的 Kudla-Rapoport 猜想，确立了特殊闭链算术相交数与修正导出局部密度的精确等式，并据此放宽了酉 Shimura 簇算术 Siegel-Weil 公式的局部假设。
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。
Normality of Schubert varieties in affine Grassmannians II: The tamely ramified case
本文研究扭曲仿射格拉斯曼流形中 Schubert 簇的正规性，利用 Levi 子群代数基本群的阶建立判据，并将其推广到等特征与混合特征下的 Pappas-Zhu 局部模型分类。
Motivic realization of rigid G-local systems on curves and tamely ramified geometric Langlands
本文证明了具有拟单值单值性和有限阶阿贝尔化的复不可约刚性 G-局部系统是动形的，推广了 Katz 的定理，并在 de Rham 几何 Langlands 设定下证明了 Hecke 特征层的存在性、谱分解及刚性与自守刚性的关系。
On the Rapoport-Zink space for $\mathrm{GU}(2, 4)$ over a ramified prime
本文研究了分歧素数上 GU(2,4) 对应 Shimura 簇的超奇异轨迹，证明了相关 Rapoport-Zink 空间的平坦性，并显式描述了基本轨迹模 p 约化的不可约分支，将其与辛群和正交群的 Deligne-Lusztig 簇联系起来。
The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.
We compute the ramification filtration on wildly ramified $p^2$-cyclic extensions of local fields of characteristic $p$. The ramification filtration on the compositum of two $p$-cyclic and $p^2$-cyclic extensions are also computed. As an application, some partial results towards Abhyankar's Inertia conjecture has been proved.
The Second Vanishing Theorem in Ramified Mixed Characteristic
在正则局部分岔混合特征标环上证明了局部上同调模的第二消没定理，并给出若干应用。
The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(ϕ, Γ_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.
In this article, we improve our main results from \emph{Chow groups and $L$-derivatives of automorphic motives for unitary groups} in two direction: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla--Rapoport conjecture for exotic smooth Rapoport--Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomolo…
The Brauer-Manin obstruction is used to explain the failure of the local-global principle for algebraic varieties. In 1999 Skorobogatov gave the first example of a variety that does not satisfy the local-global principle which is not explained by the Brauer-Manin obstruction. He did so by applying the Brauer-Manin obstruction to étale covers of the variety, and thus defining a finer obstruction. In 2008 Poonen gave the first example of failure of the local-global principle which cannot be explained for by Skorobogatov's étale-Brauer obstruction. However, Poonen's construction was not accompani…
Iwahori Fundamental Local Equivalence
该研究构造了三个具平分歧的因式分解模范畴的局部等价，含两类点处等价的因式分解版本及一类具岩堀分歧的等价，丰富了局部等价相关理论。
Singularities of local models
本文构造了Shimura簇的局部模型，证明了除剩余特征2下的奇酉群外其特殊纤维的既约性、Cohen-Macaulay性质及等特征下的（伪）有理性，并基于parahoric群概形与完美几何分析了野分歧情形下的奇点。
In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.
This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non Archimedean field $F$ of characteristic $p\neq 2$ in the framework of the geometric Langlands program. First we construct and study the geometric version of the invariants of the Weil representation of the Iwahori-Hecke algebras. In the particular case of $(GL_1, GL_m)$ we give a complete geometric description of the corresponding category. The second part of the paper deals with geometric local Langlands functoriality at the Iwahori level in a general setting. Giv…
Consider a formal (mixed-characteristic) deformation functor D of a representation of a finite group G as automorphisms of a power series ring k[[t]] over a perfect field k of positive characteristic. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. We prove that the only such D that are not pro-representable occur if k has characteristic two and G is o…
The integral motivic Satake equivalence for ramified groups
本文构建了非阿基米德局部域上准分裂reductive群的几何萨塔克等价式，扩展了Gaussent-Littelmann的工作，并推广了Zhu的积分萨塔克同构到分裂群，定义了通用球面海格尔代数并构造了通用萨塔克和伯恩斯坦同构。
To any connected reductive group G over a non-archimedean local field F (of characteristic p > 0) and to any maximal torus T of G, we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of F. This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G. For G = GL_2, we prov…
We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree $2$ over a ramified odd prime $p$ if the level at $p$ is given by a special maximal compact open subgroup. More precisely, we show that such a locus has purely $2$-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport--Zink space for the q…
Class field theory for curves over local fields
本文为局部域上的光滑射影曲线建立了分歧类域论，其证明关键在于正特征二维局部域类域论的新结果及对数 Hodge-Witt 上同调的对偶定理。
We consider the local model of a Shimura variety of PEL type, with the unitary similitudes corresponding to a ramified quadratic extension of $\mathbb{Q}_p$ as defining group. We examine the cases where the level structure at $p$ is given by a parahoric that is the stabilizer of a selfdual periodic lattice chain and that is special in the sense of Bruhat--Tits theory. We prove that in these cases the special fiber of the local model is irreducible and generically reduced; consequently, the special fiber is reduced and is normal, Frobenius split, and with only rational singularities. In additio…
Logarithmic Geometry and Geometric Class Field Theory
利用对数几何升级Deligne的几何类域论，通过定义带框架的对数Picard空间并证明除子到Picard映射的对数紧化具有对数单连通纤维，建立了曲线上的局部系统与乘法局部系统之间的典范双射，从而重新推导了函数域的适度分歧整体Artin互反律。
We consider a resolution of ramified irregular singularities of meromorphic connections on a formal disk via local Fourier transforms. A necessary and sufficient condition for an irreducible connection to have a resolution of the ramified singularity is determined as an analogy of the blowing up of plane curve singularities. We also relate the irregularity of Komatsu and Malgrange of connections to the intersection numbers and the Milnor numbers of plane curve germs. Finally, we shall define an analogue of Puiseux characteristics for connections and find an invariant of the family of connectio…
Given a finite, flat and finitely presented group scheme $G$ over some base $S$, we introduce the notion of ramified $G$-covers and study the moduli stack $G$-Cov they form. The thesis is divided in three parts. The first one concerns the case when $G$ is a diagonalizable group scheme and it essentially coincides with arxiv:1106.2347. In the second part I deal with the general case. Assuming that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers of $T$ as particolar (lax) monoidal functors from the category of finite, $G$-equivariant locally free sheaves over $S$ to the …
The Deligne--Simpson problem is an existence problem for connections with specified local behavior. Almost all previous work on this problem has restricted attention to connections with regular or unramified singularities. Recently, the authors, together with Kulkarni and Matherne, formulated a version of the Deligne--Simpson problem where certain ramified singular points are allowed and solved it for the case of Coxeter connections, i.e., connections on the Riemann sphere with a maximally ramified singularity at zero and (possibly) an additional regular singular point at infinity. A certain m…
This is an introduction to noncommutative local reciprocity maps for totally ramified Galois extensions with arithmetically profinite group. These maps in general are not homomorphisms but Galois cycles; a description of their image and kernel is included.
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。
Local types of $(Γ,G)$-bundles and parahoric group schemes
本文利用弱分歧点处稳定子群的第一非阿贝尔群上同调分类计算了光滑射影曲线上 $(Γ,G)$-丛的局部类型，并证明了特征零下泛单连通 parahoric Bruhat--Tits 群概型可由该丛导出及其局部版本。
Chow Groups: A Structure Theorem, RIEMANN-ROCH without denominators and ARTIN approximation
本文研究分歧正则局部环上的 Chow 群问题，给出该类环的刻画，证明特定开子集上 $(i-1)!\mathbb{A}^i(U)=0$，并建立 Hensel 化与完备化 Chow 群之间的构造性关系。
Author's generalization of one-dimensional class field theory to theory of abelian totally ramified p-extensions of a complete discrete valuation field with arbitrary non-separably p-closed residue field and its applications are described.
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