Ana Caraiani, Matthew Emerton, Toby Gee, David Savitt
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).
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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。
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.
It is known that for any elliptic curve $E/\mathbb{Q}$ and any integer $m$ co-prime to $30,$ the induced Galois representation $ρ_{E,m}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$ is surjective if and only if $ρ_{E,\ell}$ is surjective for any prime $\ell|m.$ In this article, we shall discuss some generalizations, applications, and variants of this phenomenon.
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
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。
For a fixed odd prime p and a representation ρof the absolute Galois group of Q into the projective group PGL(2,p), we provide the twisted modular curves whose rational points supply the quadratic Q-curves of degree N prime to p that realize ρthrough the Galois action on their p-torsion modules. The modular curve to twist is either the fiber product of the modular curves X_0(N) and X(p) or a certain quotient of Atkin-Lehner type, depending on the value of N mod p. For our purposes, a special care must be taken in fixing rational models for these modular curves and in studying their automorphis…
We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if the étale cohomology groups (with Q_ell coefficients) of the two surfaces are isomorphic as Galois representations, then the same is true of the two moduli spaces. In particular, if the field of definition is finite and the K3 surfaces have equal zeta functions, then so do the moduli spaces, even when the moduli spaces are not birational.
In this paper we investigate the cohomological obstruction for the field of moduli of a $G$-cover to be a field of definition, in the case of local fields and covers with tame admissible reduction. This applies in particular to $p$-adic fields where $p$ does not divide the order of the group $G$. We give examples of $G$-covers with field of moduli $\QQ_p$ that cannot be defined over $\QQ_p$, for all primes $p>5$.
We study the moduli space $\widetilde{\calM}$ of marked cubic surfaces. By classical work of A.\,B. Coble, this has a compactification $\widetilde{M}$, which is linearly acted upon by the group $W(E_6)$. $\widetilde{M}$ is given as the intersection of 30 cubics in $\bP^9$. For the morphism $\widetilde{calM} \to \bP(1,2,3,4,5)$ forgetting the marking, followed by Clebsch's invariant map, we give explicit formulas. I.e., Clebsch's invariants are expressed in terms of Coble's irrational invariants. As an application, we give an affirmative answer to the inverse Galois problem for cubic surfaces o…
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 show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $σ\in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^σ$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting p…
We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a $p$-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge-Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal de…
Let $L$ be a Galois algebra with Galois group $G$ and let $x$ be a normal element of $L$. The moduli space $\mathcal X$ of pairs $(L,x)$ is isomorphic to an open subset of the quotient variety $\mathbb P/G$, where $\mathbb P$ is the projective space of the regular representation of $G$. We provide a formula for the height of any pair $(L,x) \in \mathcal X(\mathbb Q)$ in terms of algebraic invariants of $L$ and $x$ with respect to a natural adelic metric on the anticanonical divisor of $\mathbb P/G$.
On the Emerton-Gee stack of potentially semistable representations
该研究利用Breuil-Kisin模变体分类潜在半稳定伽罗瓦表示，构建其栈并证明与Emerton-Gee栈同构，构造相关映射并证明其光滑性，得到对应全局正则性结果。
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.
Descent for algebraic stacks
本文研究了代数堆在fppf覆盖下的2-下降性质，通过伽罗瓦群作用重新定义了下降数据，推广了伽罗瓦下降理论。
Given an object over the algebraic closure Qbar of Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Qbar/Q), and in general this is not true, although it is sometimes surprising to observe in practice. The case of covers of the projective line branched only over the points 0, 1, and infinity, through Belyi's theorem, leads to Grothendieck's dessins d'enfants program for understanding the absolute Galois group through its faithful action on such covers. This note is motivated by Catanese's question about a higher…
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…
Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations
本文基于dagger代数的dg增强发展非阿基米德导出解析几何，为多种pro-étale层构造导出解析模函子及加权移位辛结构，并借此证明了$\ell$-进Galois表示等导出模栈上移位辛结构的存在性，证明了Minhyong Kim猜想。
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.
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 extend Kisin's results on the structure of characteristic $0$ Galois deformation rings to deformation rings of Galois representations valued in arbitrary connected reductive groups $G$. In particular, we show that such Galois deformation rings are complete intersection. In addition, we study explicitly the structure of the moduli space $X_{φ,N}$ of (framed) $(φ,N)$-modules when $G=\GL_n$. We show that when $G=\GL_3$ and $K_0=\Q_p$, $X_{φ,N}$ has a singular component, and we construct a moduli-theoretic resolution of singularities.
We prove that the non-ordinary component is connected in the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This was conjectured by Kisin. As an application to global Galois representations, we prove a theorem on the modularity comparing a deformation ring and a Hecke ring.
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.
Moduli stacks of Galois representations and the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$
本文为$\mathrm{GL}_2(\mathbb{Q}_p)$的$p$-进局部Langlands对应给出范畴论表述，将其嵌入模栈上的Ind-相干层范畴，并通过局部-整体相容公式与模曲线上同调联系。
Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two general results: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any information about a Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space.
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 …
We examine in detail the stable reduction of Galois covers of the projective line over a complete discrete valuation field of mixed characteristic (0, p), where G has a cyclic p-Sylow subgroup of order p^n. If G is further assumed to be p-solvable (i.e., G has no nonabelian simple composition factors with order divisible by p), we obtain the following consequence: Suppose f: Y --> P^1 is a three-point G-Galois cover defined over the complex numbers. Then the nth higher ramification groups above p for the upper numbering of the (Galois closure of the) extension K/Q vanish, where K is the field …
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.
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 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。
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