Deformation rings and Hecke algebras in the totally real case
K. Fujiwara
确定数域的绝对伽罗瓦群的半单l-adic表示是数论的基本问题之一。本文讨论了全实数域上的二维表示问题。
One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real number field.
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On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case
本文证明了全实域上无立方级Hecke特征的Weyl型次凸界，通过显式反演Motohashi公式并利用Schwartz函数与分布不变性。
As a continuation of previous work, we establish sum expressions for $p$-adic Hecke $L$-functions of totally real fields in the sense of Delbourgo, assuming a totally real analog of Heegner hypothesis. This is done by finding explicit formulas of the periods of the corresponding $p$-adic measures. As an application, we extend the Ferrero-Greenberg formula of derivatives of $p$-adic $L$-functions to this setting.
Cyclotomic fields are generated by cyclotomic Hecke {\it L}-values of totally real fields, II
本文研究全实域上单个临界分圆 Hecke L 值能否生成相应分圆 Hecke 域，将 Jun-Lee-Sun 的平凡驯顺特征标结果推广到可解全实域上的非平凡 Hecke 特征标。
On the theory of normalized Shintani L-function and its application to Hecke L-function
定义了多元正规化Shintani L-函数类。与Shintani zeta函数不同，归一化的Shintani L函数是全纯函数。并且满足良好的函数方程。证明了全实域上的任何Hecke L函数都可以表示为多元正规化Shintani L函数的对角部分。这给出了全实域上Hecke L函数的一个很好的多元推广。这也给出了全实域的Hecke L函数的函数方程的一个新的证明。
Arithmetic of Hecke L-functions of quadratic extensions of totally real fields
本文解决了 Shintani 方法在高维全实域上的无效性，为窄类数为1的全实域中导子为惰性有理素数的特征提供了 Shintani sets 的组合描述，简化了相关 Hecke L-函数，并将组合类数公式推广至全实域的二次扩张，有效肯定了 Hecke 猜想。
The purpose of this article is to newly define the $p$-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of $p$-adic $L$-functions interpolating nonpositive values of Hecke $L$-functions of the totally real field in terms of special values of these $p$-adic polylogarithms.
Local constancy of dimensions of Hecke eigenspaces of Automorphic forms
我们使用Buzzard的方法来研究不同类型的模形式-古典，在虚二次域和全实域的p进家庭。在偶次全实域的情形下，得到了定斜率变权空间维数的局部不变性。对于虚二次域，我们得到了与这种空间的维数上的权无关的界。
Explicit Hecke eigenform product identities for Hilbert modular forms
本文研究了Hilbert模形式的显式Hecke特征形式乘积恒等式，证明仅在Q(√5)存在两个这样的恒等式，并探讨了一般实数域的情况。
We study the Zariski closure of points in local deformation rings corresponding to potential semi-stable representations with certain prescribed $p$-adic Hodge theoretic properties. We show in favourable cases that the closure is equal to a union of irreducible components of the deformation space. We also study an analogous question for global Hecke algebras.
On isomorphisms between deformation rings and Hecke rings
我们给出了一个不同的方法，著名的Wiles和Taylor的模块提升结果。代替Taylor-Wiles系统，我们使用R的Galois上同调发现。罗摩克里希纳这篇论文比math.NT/0210296早了2年，在math.NT/0210296中提出了另一种模块化提升结果的方法。
$p$-adic $L$-functions for Hecke characters of totally imaginary fields
本文针对任意非分歧素数与全虚域，结合新提出的$p$进傅里叶理论和已有等变上同调类，构造了插值代数赫克特征临界$L$值的$p$进$L$函数，解决了此前非寻常素数情形仅适用于有理数域虚二次扩张的局限。
Hecke characters and the $K$-theory of totally real and CM number fields
设$F/K$是数域的阿贝尔扩张，$F$是CM或全实的，$K$是全实的。若$F$是CM，且Brumer-Stark猜想对$F/K$成立，我们构造了一类无穷型$F$的$G(F/K)$ --等变Hecke特征标，该特征标等于某些$G(F/K)$ --等变$L$ -函数的特殊值.利用Greither-Popescu关于Brumer-Stark猜想的结果，我们构造了这些特征的$l$ -adic非本原形式，对于素数$l> 2$。进一步，利用这些$l$ -adic Hecke特征标的特殊值，构造了$F$的$l$ -准Quillen局部化序列中的$G(F/K)$ -等变Stickelberg分裂映射，推广了Banaszak在1990年对$K = \Bbb Q$的结果.我们还应用Stickelberg分裂映射来构造$l$中的特殊元素-$K_{2n}(F)$的主块$K_{2n}(F)_l$，并分析了$K_{2n}(F)_l$中可分元素的群$D(n)_l$的伽罗瓦模结构，对于所有的$n>0$。如果$n$是奇数且与$l$互质，$F = K$是一个相当一般的全实数域，我们研究了$D(n)_l$的循环性与岩泽关于分圆域类群的经典猜想的关系，并将其推广到更广泛的数域类.最后，如果$F$是CM，我们的$l$ -adic Hecke特征标的特殊值被用来构造欧拉系统在奇数$K$ -群与系数$K_{2n+1}(F, \…
Shintani cocycles and vanishing order of $p$-adic Hecke $L$-series at $s=0$
设$χ$是全实数域$F$的有限阶Hecke特征标.利用Hill的Shintani尾环，我们给出了与$χ$相关的$p$ -adic $L$ -系列$L_p(χ, s)$的上同调结构.这用于表明$L_p(χ, s)$在$s=0$处具有平凡零，其阶至少等于$F$在$p$之上的位数，其中$χ$的局部分量是平凡的。
Rankin-Cohen Brackets of Hilbert Hecke Eigenforms
本文在窄类数一的全实域上证明希尔伯特模群两个Hecke特征形式的Rankin-Cohen括号仅因维数原因可能成为特征形式，并证明Freitag体积猜想及在维数猜想下得到乘积恒等式有限性。
Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit group. The main term can be expressed in terms of Grössencharacter $L$-functions.
Mod-$2$ Hecke algebras of level $3$ and $5$
该研究利用形变理论分析素数级模2模形式的大赫克代数，证明3、5级时其极大约化商与泛形变环的对应商同构并确定其结构，还描述了模p赫克代数的自然分次。
Deligne's conjecture on the critical values of Hecke $L$-functions
本文通过将Kings-Sprang的Eisenstein-Kronecker类解释为Blasius反射动机的de Rham类，证明了任意代数Hecke特征下Deligne关于L-函数临界值的猜想，推广了Blasius仅在CM域成立的结果。
We show that the Eigenvariety attached to Hilbert modular forms over a totally real field $F$ is smooth at the points corresponding to certain classical weight one theta series and we give a precise criterion for etaleness over the weight space at those points. In the case where the theta series has real multiplication, we construct a non-classical overconvergent generalised eigenform and compute its Fourier coefficients in terms of $p$-adic logarithms of algebraic numbers. Our approach uses deformations and pseudo-deformations of Galois representations.
Companion forms and the structure of p-adic Hecke algebras
我们研究了Hida的普通p-adic Hecke代数的Eisenstein分量的结构，它与模形式空间（mod p）中的伴随形式有关。我们证明了这样的代数是Gorenstein环，如果某些有同伴的模形式（mod p）空间是一维的，并且给出了这种一维性的一个数值判据.这与斯金纳和怀尔斯的工作部分重叠;但我们的方法，基于乌尔默的工作，是完全不同的。然后，我们考虑上述Gorenstein属性的后果。我们特别讨论了与岩泽理论的联系。
Quantum Unique Ergodicity for Eisenstein Series on the Hilbert Modular Group over a Totally Real Field
W. Luo和P.Sarnak证明了$\rm{PSL}(2,\mathbb{Z}) \backslash H$上Eisenstein级数的量子唯一遍历性。本文将他们的结果推广到$\rm{PSL}(2,O) \backslash H^n$上的Eisenstein级数，其中$O$是$Q$上的$n$度全实域上的整数环，窄类数为1。埃弗拉特本文还对非全纯Hilbert模形式上的Hecke算子理论作了简要的讨论。
Canonical Equivariant Cohomology Classes Generating Zeta Values of Totally Real Fields
本文将Dirichlet $L$-函数特殊值的经典结果推广到全实域Hecke $L$-函数，在代数环面的等变上同调中构造Shintani生成类，证明其导数在挠点处的特殊化给出Lerch型zeta函数在非正整数处的值，表明高维情形的正确框架应为等变上同调类而非函数。
Connectedness of Hecke Algebras and the Rayuela conjecture: a path to functoriality and modularity
设$ρ_1$和$ρ_2$是全实数域$F$的一对剩余的、奇的、绝对不可约的二维伽罗瓦表示.在这篇文章中，我们提出了一个猜想断言存在的“安全”链的兼容系统的伽罗瓦表示连接$ρ_1$到$ρ_2$。这样的猜想意味着广义塞尔猜想，并相当于塞尔猜想下的一个模块化版本it.We证明了一个弱版本的模块化变体使用某些Hecke代数的连通性，我们评论这些结果可能的应用，以建立一些情况下的Langlands函。
Under mild hypotheses on the residual representation, we prove the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras using a novel combination of the methods of Euler systems and Taylor-Wiles systems. We also prove the compatibility of this conjecture with specialization.
Let $F/E$ be a Galois extension of totally real number fields, with Galois group $\mathrm{Gal}(F/E)$. Let $\mathfrak{N}$ be an integral ideal which is $\mathrm{Gal}(F/E)$-invariant, and $k \ge 2$ an integer. In this note, we study the action of $\mathrm{Gal}(F/E)$ on the Hecke orbits of Hilbert newforms of level $\mathfrak{N}$ and weight $k$. We also discuss the geometric counterpart to this action, which is closely related to the notion of abelian varieties potentially of $\mathrm{GL}_2$-type. The two actions have some consequences in relation with Langlands Functoriality. We conclude with an…
Constant terms of Eisenstein series over a totally real field
本文计算了定义在全实数域上的Eisenstein级数在各种尖点处的常数项。M. Ohta计算了有理数域上权为2的Eisenstein级数在所有尖点等价类上的常数项。对于定义在全实数域上的爱森斯坦级数，S。达斯古普塔湾Darmon和R. Pollack在2011年计算了特定（不是所有）尖点等价类的常数项。本文计算了一般全实数域上Eisenstein级数在尖点的所有等价类上的常数项，并用Hecke $L$ -函数明确地描述了它们.本研究是由M.太田的工作同余模有关爱森斯坦系列定义的领域内的有理数。
Nonvanishing of special $L$-values of cusp forms on $\mathrm{GL}(2)$ with totally split prime power twists: without the class number hypothesis
本文证明在完全分裂素数幂扭下，GL(2) 上尖点形式的特殊 L 值在中心临界点非零，无需类数假设，改进了 Kwon 和 Sun 的结果。
Central values of L-functions over CM fields
本文证明了一类Rankin L函数的中心值的一个显式公式。这些L函数是全实域F上的Hilbert新形式的L函数，被F的全虚二次扩张的酉Hecke特征标扭曲。该公式推广了我们以前关于被有限特征标扭曲的L函数的结果。它还消除了对波帕论文衍生物的限制。
Unramifiedness of weight one Hilbert Hecke algebras
该研究证明了全实数域上平行权1、级与p互素的GL(2)模pⁿ尖点赫克代数的伽罗瓦伪表示在p上位非分歧，还借助Reduzzi--Xiao广义哈塞不变量构造广义Θ-算子并得到单射性判据。
We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the Langlands correspondence, i.e. on moduli spaces of Galois representations. We verify the simplest form of derived local-global compatibility between the action of the spectral Hecke algebra on the derived Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh's derived Hecke algebra on the cohomology of arithmetic groups.
We generalize a result of Galatius and Venkatesh which relates the graded module of cohomology of locally symmetric spaces to the graded homotopy ring of the derived Galois deformation rings, by removing certain assumptions, and in particular by allowing congruences inside the localized Hecke algebra.
Mass Equidistribution for Automorphic Forms of Cohomological Type on GL_2
推广了Holowinsky和Soundararajan关于SL（2，Z）上全纯Hecke模形式量子唯一遍历性的证明，建立了GL 2上任意数域上满足Ramanujan界的上同调型自守形式的量子唯一遍历性.特别地，我们得到了全实二次域和全虚二次域上的无条件定理。在全实情形下，我们证明了我们的结果蕴涵了全纯Hecke模形式的零因子的等分布，推广了Rudnick在Q上的一个结果.
Computing automorphic forms on Shimura curves over fields with arbitrary class number
本文推广了Greenberg和作者的方法，计算了定义在全实数域上的具有任意类数的Shimura曲线的上同调。通过Jacquet-Langlands对应，我们从而计算系统的Hecke特征值相关联的希尔伯特模形式的任意水平上的全实域的奇数度。最后，我们用两个例子来说明我们的算法的有效性。
On the Ramification of Hecke Algebras at Eisenstein Primes
利用Wiles的模技巧，研究了权为2、素级为N的Hecke代数在Eisenstein素数上的局部化.在此基础上，我们从形变理论的角度恢复了Mazur的一些结果（“模曲线和Eisenstein理想”）。结合Merel的一个定理，我们得到了Q（N^（1/p））的p-部分的新信息，其中p和N是素数，N = 1 mod p.
Given a continuous, odd, reducible and semi-simple $2$-dimensional representation $\barρ_0$ of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$, we study the relation between the universal deformation ring of the pseudo-representation corresponding to $\barρ_0$ (pseudo-deformation ring) and the big $p$-adic Hecke algebra to prove that the maximal reduced quotient of the pseudo-deformation ring is isomorphic to the local component of the big $p$-adic Hecke algebra corresponding to $\barρ_0$ if a certain global Galois cohomology group has dimension $1$. This partially extends th…
Ihara's lemma for Shimura curves over totally real fields via patching
本文利用拼接方法，在大像假设下证明了全实域上志村曲线模 $l$ 上同调的伊原引理，避免了 Diamond-Taylor 方法中对 $l$ 的额外限制。
In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation …
On the $I(1)$-invariants: Non-abelian Hecke algebra case
本文通过Iwahori-Hecke模型明确描述了universal module π_r的pro-p-Iwahori不变量，并确定了其作用，进而通过I(1)-不变量 functorially 恢复π_r并扩展了Ollivier的定理。
Sommes de Dedekind associées à un corps de nombres totalement réel
定义了一类附着在类数为1的全实数域上的Dedekind和.我们证明了它们满足某种互反律。然后我们将它们与Hecke $L$ -函数的特殊值联系起来。我们的结论是，他们是由斯塔克猜想。
Let $p$ be an odd prime and $f$ be a nearly ordinary Hilbert modular Hecke eigenform defined over a totally real field $F$. Let $\mathbb{I}$ be an irreducible component of the universal nearly ordinary or locally cyclotomic deformation of the representation of $\mathrm{Gal}_F$ that is associated to $f$. We study the deformation rings over a $p$-adic Lie extension $F_\infty$ that contains the cyclotomic $\mathbb{Z}_p$-extension of $F$. More precisely, we prove a control theorem about these rings. We introduce a category $\mathfrak{M}_{\mathcal H}^{\mathbb I}(\mathcal G)$, where $\mathcal G=\mat…
Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $Γ_0(\ell)$ at the maximal Eisenstein ideal containing $p$. We give a necessary and sufficient condition for the $\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for …
Eisenstein Deformation Rings
证明了某些可约剩余伽罗瓦表示的R = T定理。我们回答了格罗斯和鲁宾关于某些Hecke代数T是否是离散赋值环的一个问题。为了证明这些结果，我们（利用Breuil模的理论）确定了任意光滑分歧的离散赋值环上的两个p阶有限平坦群格式G和H在什么情况下允许一个不被p杀死的扩张.
Generation of cyclotomic Hecke fields by $L$-values of cusp forms on $\mathrm{GL}(2)$ with certain $\mathbb{Z}_p$ twist
本文证明在数域 F 上 GL(2) 的代数自守新形式 f 可由其被特定 Z_p-扩张的 Galois 特征扭曲的 L-值唯一确定，并在全实或 CM 域情形下证明 Hecke 域与分圆域的合成由这些代数 L-值生成。
On the pro-modularity in the residually reducible case for some totally real fields
本文在剩余可约情形下研究万有变形环与大 Hecke 代数的关系，证明了一个 pro-模性结果，并在某些全实域上得到条件性的大 $R=\mathbb{T}$ 定理，推广了 Deo 的结果。
Galois representations and torsion in the coherent cohomology of Hilbert modular varieties
设$F$是全实数域，$p$是$F$中的素非分歧.证明了$p$的水平互素的$F$的Hilbert模簇的相干上同调中出现的分权模$p^m$ Hecke本征类的Galois伪表示的存在性.
Mod $\ell$ non-vanishing of self-dual Hecke $L$-values over CM fields and applications
研究CM域上自对偶海克勒L-函数在模ℓ下的非消失性，并应用于Eisenstein同余可除性向CM伊瓦索主猜想的证明。
Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_\mathfrak{p}$ acting on $(\mathrm{mod}\, p^m)$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of F. Calegari and D. Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight ${\bf 1}$ are unramified at $p$ when $[F:\mathbb Q]=2$. Some pa…
For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $τ$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $τ$ is introduced by Barthel and Livné and this is defined in terms of the spherical Hecke operator. In earlier work of Anandavardhanan-Borisagar, an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient $π$ of $τ$, making use o…
Hybrid bounds for automorphic forms on ellipsoids over number fields
证明了算术型黎曼流形上Hecke-Laplace特征函数的上界。所考虑的流形是2-球面或3-球面的d-重积，实现为全实数域上四元数代数的直积。
The main result of this text is a generalization of Perrin-Riou's p-adic Gross-Zagier formula to the case of Shimura curves over totally real fields. Let $F$ be a totally real field. Let $f$ be a Hilbert modular form over $F$ of parallel weight $2$, which is a new form and is ordinary at $p$. Let $E$ be a totally imaginary quadratic extension of $F$ of discriminant prime to $p$ and to the conductor of $f$. We may construct a $p$-adic $L$ function that interpolates special values of the complex $L$ functions associated to $f$, $E$ and finite order Hecke characters of $E$. The $p$-adic Gross-Zag…
We study the local behavior of special cycles on Shimura varieties for $\mathbf{U}(2, 1) \times \mathbf{U}(1, 1)$ in the setting of the Gan-Gross-Prasad conjectures at primes $τ$ of the totally real field of definition of the unitary spaces which are split in the corresponding totally imaginary quadratic extension. We establish a local formula for their fields of definition, and prove a distribution relation between the Galois and Hecke actions on them. This complements work of \cite{jetchev:unitary} at inert primes, where the combinatorics of the formulas are reduced to calculations on the Br…
Bost-Connes systems, Hecke algebras, and induction
该研究引入与数域上全正仿射群关联的Hecke代数，证明其是Bost-Connes C*-代数的满角并得到相变定理，还将Bost-Connes系统构造扩展为数域到C*-动力系统的函子并研究了KMS态的诱导性质。
Companion Forms Over Totally Real Fields, II
证明了模Hilbert模形式的一个伴随形式定理.本文推广了Gross和Coleman-Voloch关于Q上模形式的结果，并在许多情况下给出了他们结果的新证明.所使用的方法是完全不同的，以前的工作在这方面，并依赖于模块提升定理和变形的伽罗瓦表示的一般理论。
We prove the existence of a potentially diagonalizable lift of a given automorphic mod $p$ Galois representation $\overlineρ:{\rm Gal}(\overline{F}/F)\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ for any totally real field $F$ and any rational prime $p>2$ under the adequacy condition by using automorphic lifting techniques developed by Barnet-Lamb, Gee, Geraghty, and Taylor. As an application, when $p$ is split completely in $F$, we prove a variant of Serre's weight conjecture for $\overlineρ$. The formulation of our Serre conjecture is done by following Toby Gee's philosophy. Applying…
We continue our study of the Wiles defect of deformation rings $R$ and Hecke rings $T$ (at a newform $f$) acting on the cohomology of Shimura curves. The Wiles defect at an augmentation $λ_f:T \to O$ measures the failure of $R,T$ to be complete intersections locally at $λ_f$. In situations we study here the Taylor-Wiles-Kisin patching method gives an isomorphism $R=T$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of our earlier results that compute the Wiles defect at $λ_f: R=T \t…
Finiteness of unramified deformation rings
证明了连续伽罗瓦表示$\overlineρ: G_{F^{+}} \rightarrow \mathrm{GL}_n(k)$（全实数域$F^{+}$和有限域$k$）的泛非分歧变形环$R^{\mathrm{unr}}$在许多情况下在$\mathcal{O} = W(k)$上是有限的.我们还证明了（在类似的假设下），通用变形环$R^{\mathrm{univ}}$是有限的局部变形环$R^{\mathrm{loc}}$。
The Hodge Realization of the Polylogarithm and the Shintani Generating Class for Totally Real Fields
本文在关联全实域的代数环面等变 Deligne-Beilinson 上同调中构造多对数类的霍奇实现，证明其 de Rham 实现给出 Shintani 生成类，并提出关于挠点特殊化的猜想及其与 Beilinson 猜想的联系。
Nearly Ordinary Galois Deformations over Arbitrary Number Fields
设K是任意数域，ρ：Gal（Kbar/K）-> GL_2（E）是几乎普通的不可约几何伽罗瓦表示.本文研究了ρ的近寻常变形。当K是全实的和ρ是模块化的，Hida的结果意味着，几乎普通的变形空间与ρ包含一个Zebriki密集的点集对应于“自守”的伽罗瓦表示。我们猜想，如果K是_not_ totally real的，那么这是从来没有的情况下，除了在三个例外的情况下，对应于（1）“基地变化”，（2）“CM”形式，（3）“甚至”表示。后一种情况理论上只能在ρ的象是有限的情况下发生。我们的结果有两种。首先，我们证明了一个一般的结果阿廷表示，条件加强Leopoldt猜想。其次，当K是一个虚二次域，我们证明了一个无条件的结果，意味着存在“许多”正维组件（某些变形空间），不包含无限多的经典点。还包括一些关于“p-adic函数性”的推测性评论，以及一些关于当n > 2时我们的方法如何应用于Gal（Qbar/Q）的n维表示的评论。
We study in this paper Hida's p-adic Hecke algebra for GL_n over a CM field F. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case of F being the rationals implies the classical Leopoldt conjecture for a number field K of degree n over the rationals, if one assumes further the existence of automorphic induction of characters for the extension K over the rationals. We study Hida's conjecture using the automorphy lifting techniques adapted to the GL_n setting by Calegari--Geraghty. W…
The Hecke algebra action on Morava E-theory of height 2
给定一个高度为2的一维形式群，设E是Lubin-Tate环上与其万有形变相关的Morava E-理论谱。通过对椭圆曲线模空间的计算，给出了作用于E-上同调的Hecke算子代数的一个显式.这导致了Rezk的对数上同调运算在E的单位上的零结果。它确定了一族亚纯模形式的Serre导数为零的核心元素。我们的计算发现了与模数单位的矩阵的联系。特别地，我们在Knopp和Mason的意义下给出了Hecke算子在某些“对数”q-级数上的作用，它与我们的消失结果一致，并且推广了经典的Hecke算子在模形式上的作用.
Let $F$ be a CM field and let $(\overline{r}_{π,λ})_λ$ be the compatible system of residual $\mathcal{G}_n$-valued representations of $\operatorname{Gal}_{F}$ attached to a RACSDC automorphic representation $π$ of $\operatorname{GL}_n(\mathbb{A})$, as studied by Clozel, Harris and Taylor and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to $\overline{r}_{π,λ}$ are unobstructed for all places $λ$ in a subset of Dirichlet density $1$, continuing the investigations of Mazur, Weston and Gamzon. During the proof, we develop a general framew…
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