Matthew Emerton, Toby Gee
This survey article is based on 10 lectures that we gave at the September 2019 Hausdorff School in Bonn on "the Emerton--Gee stack and related topics", and the sections correspond to the original lectures. We have retained the somewhat informal style of some of these lectures, but have filled them out with further details.
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We construct stacks which algebraize Mazur's formal deformation rings of local Galois representations. More precisely, we construct Noetherian formal algebraic stacks over Spf Zp which parameterize étale (phi,Gamma)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. We use these stacks to show that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero, and indeed admit crystalline lifts. We also discuss the relationship between the geometr…
We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, as well as their resolutions by moduli stacks of two-dimensional Breuil-Kisin modules with tame descent data. We study the local geometry of these moduli stacks by comparing them with local models of Shimura varieties at hyperspecial and Iwahori level.
Moduli stacks of generalized phi-modules
该研究改进了Emerton-Gee堆的群变换代表性结论，展示了其在p进域上的有限呈现代数堆结构。
We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).
In this article we study motives corresponding to the moduli stacks of G-shtukas and their local models. In particular we deal with the question of describing their motivic fundamental invariants. As an application, we provide a criterion for mixed Tateness of the local model, and discuss the semi-simplicity of Frobenius on their cohomology. We then use the theory of local models to reformulate a purity result for these moduli stacks in the motivic context.
We define and study stacks which parametrize Lubin--Tate $(φ,Γ)$-modules. By working at a perfectoid level, we compare these with the Emerton--Gee stacks of cyclotomic $(φ,Γ)$-modules. As a consequence, we deduce perfectness of the Herr complex in the Lubin--Tate setting.
In this article we develop the theory of local models for the moduli stacks of global $G$-shtukas, the function field analogs for Shimura varieties. Here $G$ is a smooth affine group scheme over a smooth projective curve. As the first approach, we relate the local geometry of these moduli stacks to the geometry of Schubert varieties inside global affine Grassmannian, only by means of global methods. Alternatively, our second approach uses the relation between the deformation theory of global $G$-shtukas and associated local $P$-shtukas at certain characteristic places. Regarding the analogy be…
Components of moduli stacks of two-dimensional Galois representations
本文研究此前引入的二维驯顺潜在 Barsotti-Tate 表示模栈及相关 Breuil-Kisin 模模栈，刻画其不可约分支，并证明表示模栈的分支由特定 Serre 权重自然索引。
On the variation of slopes over moduli stacks of Galois representations
本文研究了模形式p进斜率随伽罗瓦表示模p变化的关系，并证明了幽灵猜想在直和下具有稳定性。
Let phi: P^1 --> P^1 be a rational map defined over a field K. We construct the moduli space M_d(N) parameterizing conjugacy classes of degree-d maps with a point of formal period N and present an algebraic proof that M_2(N) is geometrically irreducible for N>1. Restricting ourselves to maps phi of arbitrary degree d >= 2 such that the composition h^{-1} phi h = phi for some nontrivial h in PGL_2, we show that the moduli space parameterizing these maps with a point of formal period N is geometrically reducible for infinitely many N.
This is the second in a sequence of articles, in which we explore moduli stacks of global G-shtukas, the function field analogs for Shimura varieties. Here G is a flat affine group scheme of finite type over a smooth projective curve C over a finite field. Global G-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global G-shtukas are algebraic Deligne-Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands…
The Moduli Stack of Breuil-Kisin Modules with Descent Data for Reductive Groups
该研究引入并研究带$\u0120$-结构与下降数据的Breuil-Kisin模的模叠，定义了权有界的对应模叠，证明其为$p$进形式代数叠且与扭Schubert簇光滑等价，是作者博士论文的整理版本。
Arithmetic Hodge-Iwasawa Moduli Stacks
本文在已有Hodge-Iwasawa理论基础上构造G-丛的算术模叠，用以连贯参数化Hodge-Iwasawa结构。
We construct certain rational functions (modular units) on the moduli stack of Drinfeld shtukas. The divisors of these rational functions are supported on horospherical divisors of the moduli stack. The key to our construction is a vanishing theorem for shtukas with zeros and poles satisfying certain conditions. Using deformation theory, we calculate the divisors of these rational functions.
We describe the action of the different Frobenius morphisms on the cohomology ring of the moduli stack of algebraic vector bundles of fixed rank and determinant on an algebraic curve over a finite field in characteristic p and analyse special situations like vector bundles on the projective line and relations with infinite Grassmannians.
We show the compactly supported motive of the moduli stack of degree $n$ rational curves on the weighted projective stack $\mathcal{P}(a,b)$ is of mixed Tate type over any base field $K$ with $\text{char}(K) \nmid a,b$ and has class $\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}$ in the Grothendieck ring of stacks. In particular, this improves upon the result of [HP] regarding the arithmetic invariant of the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a …
Root stack valuative criterion for good moduli spaces
本文在剩余特征的温和假设下，证明了好模空间映射与约化群gerbe的根叠赋值判别法，并给出了其在挠子parahoric扩张、叠有理点、gerbe与齐性空间及纤维丛几何中的应用。
In this note we intend to look at the moduli stacks for global $G$-shtukas from a new perspective. We discuss a unifying interpretation of several moduli spaces (stacks) including moduli of global $G$-shtukas and (a variant of the) moduli of Higgs bundles. We view these spaces (stacks) as different fibers of a family over a scheme (stack) locally of finite type. We discuss (a relative version of) the local model theory for this family. We also consider the Hecke stacks over the moduli stack of $G$-shtukas and discuss the corresponding (motivic) Hecke operations.
In the study of automorphic representations over a function field, Hitchin moduli stack and its variants naturally appear and their geometry helps the comparison of trace formulae. We give a survey on applications of this observation to a relative fundamental lemma, the arithmetic fundamental lemma and to the higher Gross-Zagier formula.
An abelian category of relative pure motives is constructed along the lines of André (over a field of characteristic 0). An algebraic stack is shown to possess a motive in this sense. This motive is studied for the moduli stack of G-bundles over the universal curve for appropriate G, and a motivic Atiyah-Bott formula is obtained. By taking Hodge realization, a description of the corresponding variations of Hodge structures over M_g is given. In particular, it follows that these lie in the tensor category generated by the variation of Hodge structure associated to the curve itself. Similar resu…
Cohomology theories in the moduli of ring stacks
本文证明了环的合成化到代数栈的映射是全忠实的，回答了 Drinfeld 的问题，并在多种上同调理论中给出类似结果。
Arithmetic volumes of moduli stacks of Shtukas
本文研究模堆栈中tautological classes的算术体积及其与Artin L函数导数的关系，并探讨phantom tautological ring的结构及其在函数域Colmez猜想中的应用。
Moduli stacks of crystals and isocrystals
本文构造了晶体与等晶体的模叠，证明其分别为形式代数叠和 adic 叠，并利用 Verschiebung 不动点几何计数秩一 F-等晶体。
Murphy's Law for Algebraic Stacks
该研究证明曲线叠等多种自然代数几何模叠满足“代数叠墨菲定律”，即域上所有Deligne-Mumford gerbe都可作为其点的剩余gerbe，对应对象以所有可能方式无法在模域定义、粗模空间无法成为精细模空间，核心依赖等变射影几何构造向相对几何的推广。
We compute the Brauer group of the moduli stack of elliptic curves over the integers, localizations of the integers, finite fields of odd characteristic, and algebraically closed fields of characteristic not $2$. The methods involved include the use of the parameter space of Legendre curves and the moduli stack of curves with full (naive) level $2$ structure, the study of the descent spectral sequence in étale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of $S_3$ in a certain integral representation, the classification of cubic Galois e…
Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $\G$ be a connected semisimple group scheme over $X$. Under certain hypothesis we prove the equality of two numbers associated with $\G$. The first is an arithmetic invariant, its Tamagawa number. The second, is a geometric invariant, the number of connected components of the moduli stack of $\G$-torsors on $X$.
A relation between zip stacks and moduli stacks of truncated local shtukas
本文证明了 Viehmann 双陪集空间由 Lusztig 簇表示，从而实现了截断局部 shtuka 模栈与 zip 栈的比较，并在正特征完美域上建立了粗模栈间的同胚，深化了对 Shimura 簇中 zip 周期映射的理解。
For a split reductive group G over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of l in the Langlands correspondence for function fields.
Schwartz $κ$-densities for the moduli stack of rank $2$ bundles on a curve over a local field
本文研究局部域上曲线上秩2向量丛模栈，建立Schwartz $κ$-密度与极稳定向量丛粗模空间光滑函数的关联，并在非阿基米德情形证明其$κ$-有界性。
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