过收敛Hilbert特征形的经典性：二次剩余度情形

Partial Classicality of Hilbert Modular Forms

Let F be a totally real field and p a rational prime unramified in F. We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to certain but not all weights, an overconvergent form is partially classical. We use the method of analytic continuation.

Slope classicality via completed cohomology

We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of $\mathrm{GL}_2(\mathbb{Q}_p)$. The $U_p$ operator becomes a double-coset, and unitarity yields the slope vanishing.

2-adic特征曲线是真的

The 2-adic Eigencurve is Proper

当p=2，驯服水平N=1时，我们证明了从（Coleman-Mazur）特征曲线到权空间的映射满足适定性的赋值准则.更非正式地，我们表明，特征曲线没有“洞”，给定一个穿孔的圆盘的有限斜率过收敛的特征形在权空间，中心可以“填补”与有限斜率过收敛的特征形。

过收敛模函数的谱展开

Spectral expansions of overconvergent modular functions

本文的主要结果是Gouvea和Mazur（Math.Res.Lett.，1995），它断言对于特定的r值，tame层N和权k的r-过收敛p-adic模形式的空间应该被有限斜率Hecke特征形所张成。使用改编自Buzzard和Calegari工作的方法，我证明了对于p=2，k = 0，N = 1，这对（5/12，7/12）中的所有r都成立。证明依赖于一定的因式分解的U_p算子，这是已知的，在这种情况下，但我猜想也适用于p = 3和5;这一猜想也意味着确切的公式为一组类似的斜率证明p=2的Buzzard和Calegari。同样的方法也提供了一个有效的方法来显式计算小斜率过收敛特征形的q-展开式，扩展了Gouvea和Mazur的计算。

Parabolic eigenvarieties via overconvergent cohomology

Let $G'$ be a connected reductive group over $\mathbb{Q}$ such that $G = G'/\mathbb{Q}_p$ is quasi-split, and let $Q \subset G$ be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to $Q$, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When $Q$ is the Borel, we recover the …

On locally analytic vectors of the completed cohomology of modular curves

We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight $k$, we show its corresponding Galois representation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.

特征簇上的光滑性与经典性

Smoothness and Classicality on eigenvarieties

本文利用“修补特征簇”，在任意维数下证明了定酉群上过收敛 p-adic 自守形式在经典权且局部 Galois 表示为晶体时成为经典自守形式的新情形。

L-Indistinguishability on Eigenvarieties

In this article we construct examples of L-indistinguishable overconvergent eigenforms for an inner form of SL(2).

Siegel模尖形的P-Adic族

P-Adic families of Siegel modular cuspforms

设p是奇素数，g是大于或等于2的整数.证明了亏格为g的有限斜率Siegel尖点特征形在g维权空间上可以p-向变形.这个结果的证明依赖于构造一族局部解析过收敛模形式的层。

p-adic Hilbert模形式的升阶

Level raising for p-adic Hilbert modular forms

本文将作者以前的工作推广到全实数域上定四元数代数的过收敛$p$ -adic自守形式的设定。我们证明的结果是类似于经典的“水平提升”的结果，在理论的mod $p$模形式。粗略地说，我们表明，一个过收敛的本征形，其相关的局部伽罗瓦表示在某些辅助素数$ł$是（扭曲）的直和平凡和分圆字符在于一个家庭的本征形的局部伽罗瓦表示在$ł$一般是（扭曲）的分歧扩展平凡的分圆。我们给出了一些明确的例子，$p$ -adic自守形式，我们的结果适用，并给出了一个一般家庭的例子，其存在性将遵循从反例的Leopoldt猜想全实领域。这些结果也起到了技术性的作用，在其他工作的作者的问题的局部-整体相容性的Steinberg地方的Hilbert模形式的部分重量1。

希尔伯特模形式的经典性

Classicité de formes modulaires de Hilbert

本文证明了过收敛Hilbert模形式的一个经典结果。为了得到这个结果，我们使用了解析延拓方法，该方法首先由Buzzard和Kassaei使用。我们证明了这个结果没有任何分歧的假设。

Lubin-Tate theory and overconvergent Hilbert modular forms of low weight

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $Γ$ be the Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a classification of $p$-adic representations of $\mathrm{Gal}\left(\overline{K}/K\right)$ in terms of $(φ,Γ)$-modules. A useful aspect of this classification is Berger's dictionary which expresses invariants coming from $p$-adic Hodge theory in terms of these $\left(φ,Γ\right)$-modules. In this paper, we use the theory of locally analytic vectors to generalize this dictionary to the setting where $Γ$ is the Galois group of a Lubin-Tate extension of $K$. …

超收敛模形式的经典性

Classicité de formes modulaires surconvergentes

本文证明了与$\mathbb{Q}_p$上的非分歧约化群相关联的（A）或（C）型PEL Shimura簇上的过收敛模形式的一个经典性结果.为了得到这个结果，我们使用了解析延拓方法，该方法首先由Buzzard和Kassaei使用。

Formes modulaires surconvergentes, ramification et classicité

We prove in this paper a classicality result for overconvergent modular forms on PEL Shimura varieties of type (A) or (C), without any ramification hypothesis. We use an analytic continuation method, which generalizes previous results in the unramified setting. We work with the rational model of the Shimura variety, and use an embedding into the Siegel variety to define the integral structures on the rigid space.

Les Variétés de Hecke-Hilbert aux points classiques de poids $1$

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.

Reductions of Galois representations and the theta operator

Let $p\ge 5$ be a prime, and let $f$ be a cuspidal eigenform of weight at least $2$ and level coprime to $p$ of finite slope $α$. Let $\barρ_f$ denote the mod $p$ Galois representation associated with $f$ and $ω$ the mod $p$ cyclotomic character. Under an assumption on the weight of $f$, we prove that there exists a cuspidal eigenform $g$ of weight at least $2$ and level coprime to $p$ of slope $α+1$ such that $$\barρ_f \otimes ω\simeq \barρ_g,$$ up to semisimplification. The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions…

Theta 运算符等于 Fontaine 运算符在模曲线上的

Theta Operator Equals Fontaine Operator on Modular Curves

本文证明了在模曲线中，theta运算符与Fontaine运算符相等的条件，并展示了该等价关系与模形式的经典性及伽罗瓦表示的de Rham性质之间的联系。

具有稠密普通轨迹的PEL型Shimura簇上尖形的特征簇

Eigenvarieties for cuspforms over PEL type Shimura varieties with dense ordinary locus

设p>2是素数，X是（A）或（C）型紧化的PEL Shimura簇，使得p是PEL数据的非分歧素数.利用Andreatta，Iovita，Pilloni和Stevens的几何方法，我们定义了X的Iwahoric水平的过收敛局部解析p-adic模形式族的概念.我们证明了Iwahoric水平的任何有限斜率尖点本征形的本征值系统都可以变形为一族生活在权空间的开子集上的本征值系统。为了证明这些结果，我们实际上构造了期望维数的本征簇，该本征簇参数化出现在尖点形式族空间中的本征值系统。