Christian Johansson, Jack A. Thorne
We study the conditions imposed by conjectures of Arthur and Kottwitz on the Galois representations occurring in the cohomology of Shimura varieties.
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Let $A$ be an abelian variety over a number field $K$ with good reduction outside a finite set of primes $S$. We show that if the $\ell$-torsion subgroup schemes $A[\ell^n]$ lie in a certain category of group schemes, then $A[\ell^n]$ does not contain any subgroup schemes that are étale or are of multiplicative type.
Semisimplicity of étale cohomology of certain Shimura varieties
本文在假设相关自守伽罗瓦表示存在且满足良好性质的前提下，利用Fayad-Nekovář的抽象半单性判别准则与Eichler-Shimura关系，证明了某些阿贝尔型志村簇的étale上同调的特定部分具有半单性。
We give a new proof of Ohta's Lambda-adic Eichler-Shimura isomorphism using p-adic Hodge theory and the results of Bloch-Kato and Hyodo on p-adic etale cohomology.
We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, embeds into the reduction modulo p of a semistable Galois representation with Hodge-Tate weights in the expected range (at least after semisimplifying, in the case of the cohomological degree > 1). We prove refinements with descent data, and we apply these results to the cohomology of unitary Shimura varieties, deducing vanishing results and applications to the weight part of Serre's conjecture.
Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction…
Integral models of Shimura varieties with parahoric level structure, II
本文构造了具有paraoric层级结构的Abelian型Shimura变种的整数模型，这些模型在étale局部上与相应的局部模型同构。
In this paper, we define and construct canonical filtered $F$-crystals with $G$-structure over the integral models for Shimura varieties of abelian type at hyperspecial level defined by Kisin. We check that these are related by $p$-adic comparison theorems to the usual lisse sheaves, and as an application we also use this to show that the Galois representations generated from the $p$-adic étale cohomology of Shimura varieties with nontrivial coefficient sheaves are crystalline, at least in the case of proper abelian type Shimura varieties.
I construct a generalisation of Mantovan's almost product structure to Shimura varieties of Hodge type with hyperspecial level structure at $p$ and deduce that the perfection of the Newton strata are pro-étale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani's and Scholze's generalisation of the almost product structure for Shimura varieties of Hodge type.
On the étale cohomology of Hilbert modular varieties with torsion coefficients
本文研究带挠系数的希尔伯特模簇平展上同调，借鉴酉志村簇方法得到一般情形下的中间次数集中零化定理与非一般情形的次数界，借助几何Jacquet--Langlands函子性证明并应用于p进局部朗兰兹对应。
We describe work of Faltings on the construction of étale cohomology classes associated to symplectic Shimura varieties and show that they satisfy certain trace compatibilities similar to the ones of Siegel units in the modular curve case. Starting from those, we construct a two variable family of trace-compatible classes in the cohomology of a unitary Shimura variety.
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.
We study $p$-adic integral models of certain PEL Shimura varieties with level subgroup at $p$ related to the $Γ_1(p)$-level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated with unitary groups that split over an unramified extension of $\mathbb{Q}_p$ and the case of Siegel modular varieties. We construct local models, i.e. simpler schemes which are étale locally isomorphic to the integral models. Our integral models are defined by a moduli scheme using the notion of an Oort-Tate generator of a group scheme. We use these local models t…
The absolute Galois group of the cyclotomic field $K={\mathbb Q}(ζ_p)$ acts on the étale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for $K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of $K$. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology, which determin…
Compatibility of $F$-isocrystals on adjoint Shimura varieties
研究伴随志村簇上典范\(\ell -\)进局部系和典范\(F -\)等晶体的相容性，方法依赖典范\(p -\)进局部系晶体性、Margulis超刚性及相关构造，扩展了前两位作者过去的结果。
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport \cite{LR87} of deriving the Kottwitz's formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands-Rapoport conjecture), which replaces their Galois gerb theoretic arguments by more geometric …
Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $ρ$ be a $2$-dimensional $p$-adic representation of $\mathrm{Gal}(\bar{F}/F)$ which appears in the étale cohomology of quaternion Shimura curves (thus $ρ$ is associated to Hilbert eigenforms). When the restriction $ρ_{\wp}:=ρ|_{D_{\wp}}$ at the decomposition group of $\wp$ is semi-stable non-crystalline, one can associate to $ρ_{\wp}$ the so-called Fontaine-Mazur $\mathcal{L}$-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these $\mathcal{L}…
Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I
该研究证明了任意非分歧混合特征(0,p)下Hodge型志村簇良好光滑整模型的存在性，给出Langlands猜想的光滑解，并推进了Milne和Reimann相关猜想的证明，证明方法新颖且更具原理性。
Let G be a unitary group over a totally real field, and X a Shimura variety for G. For certain primes p of good reduction for X, we construct cycles on the characteristic p fiber of X. These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on X. The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group G', which is isomorphic to G at finite places but not isomorphic to G at archimedean places. More precisely, each such cycle has a natural des…
On the constituents of the mod $p$ cohomology of Shimura curves
研究在特定条件下\(\mathrm{GL}_2(K)\)的容许光滑表示\(\pi\)，通过先前工作证明其具有有限长度，本文进一步得到关于\(\pi\)子商结构的精细结果，包括岩泽基座过滤和\(K_1\)-不变量，还确定了其希尔伯特级数。
Norm-compatible families of cohomology classes for Shimura varieties, and other arithmetic symmetric spaces, play an important role in Iwasawa theory of automorphic forms. The aim of this note is to give a systematic approach to proving "vertical" norm-compatibility relations for such classes (where the level varies at a fixed prime p), treating the case of Betti and étale cohomology at once, and revealing an unexpected relation to the theory of spherical varieties. This machinery can be used to construct many new examples of norm-compatible families, potentially giving rise to new constructio…
For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke correspondence acting on the compactly supported étale cohomology and verify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization of that formula. The main ingredients of our proof of the formula are a recent work of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory of Galois gerbs developed by Langlands and Rapoport \cite{LR87}. Es…
Let G be a unitary group over the rationals, associated to a CM-field F with totally real part F^+, with signature (1,1) at all the archimedean places of F^+. Under certain hypotheses on F^+, we show that Jacquet-Langlands correspondences between certain automorphic representations of G and representations of a group G' isomorphic to G except at infinity can be realized in the cohomology of Shimura varieties attached to G and G'. We obtain these Jacquet-Langlands correspondences by studying the bad reduction of a Shimura variety X attached to G at a prime p for which X has maximal parabolic le…
The cohomology of certain intermediate strata of Kottwitz varieties
本文利用迹公式和p-进一般线性群表示，推导出Kottwitz簇某些层调的étale上同调Frobenius-Hecke迹的显式公式。
Geometry around unramified CM points
本文通过构造特殊affinoid并应用正深度Deligne-Lusztig理论，计算了Yu构造的超尖表示对局部Shimura簇中间上同调的显式贡献，并利用局部shtukas叠解决了典型子群问题。
In a work of T. Saito, the action of algebraic correspondences on the etale cohomology of varieties over local fields with semistable reduction is related to correspondences on smaller strata via weight spectral sequences. We give an intersection theoretic construction of these correspondences. Under a finiteness condition this enables us to compute them without involving the blow-ups of products, and prove their compatibility with compositions. These features are essential for the application to Shimura varieties.
This paper treats what we call `weak geometric liftings' of Galois representations associated to abelian varieties. This notion can be seen as a generalization of the idea of lifting a Galois representation along an isogeny of algebraic groups. The weaker notion only takes into account an isogeny of the derived groups and disregards the centres of the groups in question. The weakly lifted representations are required to be geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question states that any irreducible geometric representation is a twist of a subquotient of …
A de Rham weight part of Serre's conjecture and generalized mod $p$ BGG decompositions
本文通过de Rham上同调与BGG分解提出Serre猜想的几何版本，并证明其等价性，同时改进了相关结果。
Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne group of F. This bijection is required to satisfy certain natural compatibilities, of which the most important is compatibility with local functional equations (preservation of L and epsilon factors of pairs). It is enough to construct a bijection with these properties between supercuspidal representations of GL(n,F) and n-dimensional irreducible representa…
Purity for Perfectoidness
本文研究 perfectoid 性的解析局部性，建立赋值与微分准则及纯净性结果，并用于证明光滑 $p$ 进簇高阶完备 étale 上同调消失。
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