Eichler-Shimura Relation on Intersection Cohomology

完美体单位ary Shimura 变种与 $p$-adic Eichler-Shimura 映射 I

Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I

本文通过完美体构造和收敛Eichler-Shimura映射，研究单位ary群上的$p$-adic自动机形与特征种几何及$p$-adic $L$-函数的关系。

Seed Relations for Eichler--Shimura congruences and Euler systems

This paper proves that the $\mathbb{U}$-operator attached to a cocharacter is a right root of the corresponding Hecke polynomial. This result is an important ingredient in the proof of (i) the horizontal norm relations in the context of Gross--Gan--Prasad cycles and of (ii) the generalization of Eichler--Shimura relations.

Overconvergent Eichler-Shimura isomorphisms

We provide a geometric Hodge-Tate map giving generic description of the overconvergent modular symbols of some p-adic (accessible) weight k, base-changed to C_p, in terms of overconvergent modular forms of weight k+2.

p-adic Eichler-Shimura maps for the modular curve

We give a new proof of Faltings's p-adic Eichler-Shimura decomposition of the modular curves via BGG methods and the Hodge-Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was already used in the original proof. Then, we construct overconvergent Eichler-Shimura (ES) maps for the modular curves providing ''the second half'' of the overconvergent ES map of Andreatta-Iovita-Stevens. We use higher Coleman theory on the modular curve developed by Boxer-Pilloni to show that the small slope part of the ES maps interpolates the classical p-adi…

The Lambda-adic Eichler-Shimura isomorphism and p-adic etale cohomology

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.

完美化过收敛Siegel模形式与过收敛Eichler—Shimura映射

Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism

本文通过推广完美化方法构建了Siegel模形式的过收敛自守sheaf，其全局截面即为过收敛Siegel模形式，并建立了适用于Siegel模形式的过收敛Eichler—Shimura映射。

Zeta-polynomials, Hilbert polynomials, and the Eichler-Shimura identities

In 2017, Ono, Rolen, and Sprung [ORS17] answered problems of Manin [Man16] by defining zeta-polynomials $Z_f(s)$ for even weight newforms $f\in S_k(Γ_0(N)$; these polynomials can be defined by applying the "Rodriguez-Villegas transform" to the period polynomial of $f$. It is known that these zeta-polynomials satisfy a functional equation $Z_f(s) = \pm Z_f(1-s)$ and they have a conjectural arithmetic-geometric interpretation. Here, we give analogous results for a slightly larger class of polynomials which are also defined using the Rodriguez-Villegas transform.

志村簇与p进Shtuka模空间的S=T猜想

S=T for Shimura Varieties and Moduli Spaces of p-adic Shtukas

本研究借助Scholze的钻石、v-栈理论及Fargues–Scholze几何Satake等价，证明了Xiao–Zhu提出的S=T猜想，并推导出Hodge型志村簇的Eichler–Shimura关系。

尖点形式的标量值深度二 Eichler-Shimura 积分

Scalar-valued depth two Eichler-Shimura Integrals of Cusp Forms

本文定义了深度二全纯迭代 Eichler-Shimura 积分 $I_{f,g}$，构造了以其为顶部分量的向量值模形式，导出可有效计算的标量值深度二积分 $\mathcal{E}_{f,g}$，并解释了周期多项式的正交关系。

Congruence relations of GSpin Shimura varieties

In these notes I proved the Chai-Faltings version of Eichler-Shimura congruence relation for simple GSpin Shimura varieties. This extends the results by Bueltel, Wedhorn and Koskivirta.

The Eichler cohomology theorem for Jacobi forms

Let $Γ$ be a finitely generated Fuchsian group of the first kind which has at least one parabolic class. Eichler introduced a cohomology theory for Fuchsian groups, called as "Eichler cohomology theory", and established the $\CC$-linear isomorphism from the direct sum of two spaces of cusp forms on $Γ$ with the same integral weight to the Eichler cohomology group of $Γ$. After the results of Eichler, the Eichler cohomology theory was generalized in various ways. For example, these results were generalized by Knopp to the cases with arbitrary real weights. In this paper, we extend the Eichler c…

Eichler Cohomology of Generalized Modular Forms of Real Weights

In this paper, we prove the Eichler cohomology theorem of weakly parabolic generalized modular forms of real weights on subgroups of finite index in the full modular group. We explicitly establish the isomorphism for large weights by constructing the map from the space of cusp forms to the cohomology group.

On the generic part of the cohomology of local and global Shimura varieties

Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative approach to vanishing results of Caraiani-Scholze for the cohomology of unitary Shimura varieties.

Compactly supported cohomology and nearby cycle cohomology of open Shimura varieties of PEL type

In this paper, we compare two cohomology groups associated to Shimura varieties of PEL type, which are not proper over the base. One is the compactly supported l-adic cohomology, and the other is the nearby cycle cohomology, namely, the compactly supported cohomology of the nearby cycle complex for the canonical integral model of the Shimura variety over Z_p. We prove that the G(Q_p)-cuspidal part of these cohomology groups are the same, where G denotes the reductive algebraic group naturally attached to the PEL datum. Some applications to unitary Shimura varieties are also given.

带权上同调、Hodge理论与Shimura簇的交上同调

Weighted Cohomology, Hodge Theory and Intersection Cohomology of Shimura varieties

本文证明了复Shimura簇的Baily-Borel紧化交上同调与重合Hodge结构的顶权商相关，引入了与特殊循环相关的规范循环类，将非紧Shimura簇的分析几何体积与拓扑项联系起来。

某些志村簇的étale上同调的半单性

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。

Multiple period integrals and cohomology

This work gives a version of the Eichler-Shimura isomorphism with a non-abelian $H^1$ in group cohomology. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. We show Manin's map is injective but far from surjective. By extending Manin's map we are able to construct a bijective map and remarkably this establishes the existence of a non-abelian version of the Eichler-Shimura map.

Eichler cohomology in general weights using spectral theory

In this paper, we construct a pairing between modular forms of positive real weight and elements of certain Eichler cohomology groups that were introduced by Knopp in 1974. We use spectral theory of automorphic forms to show that this pairing is perfect for all positive weights except 1. The approach in this paper gives a new proof of a theorem by Knopp and Mawi from 2010 for all real weights excluding 1 and also a version of this theorem for vector-valued modular forms.

Shimura簇的相干上同调与自守形式

Coherent cohomology of Shimura varieties and automorphic forms

本文证明了Shimura簇环面紧致化上自守向量丛典范扩张的上同调可通过自守形式的相对李代数上同调计算，该结果平行于Borel和Franke的工作并推广了Mumford的定理。

Igusa叠与志村簇的上同调

Igusa Stacks and the Cohomology of Shimura Varieties

本文针对所有Hodge型志村簇构造函子性Igusa叠，依托该构造及Fargues-Scholze谱作用证明其与半单局部朗兰兹对应的相容性，还在紧合情形得到新的上同调挠消失结果。

$l$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

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…

Eichler cohomology and zeros of polynomials associated to derivatives of $L$-functions

In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case …

The Plectic Weight Filtration on Cohomology of Shimura Varieties and Partial Frobenius

We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovar and Scholl. This is achieved with the help of Morel's work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves $X_0(D, N)$ attached to an Eichler order of square-free level $N$ in an indefinite rational quaternion algebra of discriminant $D>1$. We prove that, when the genus $g$ of the curve is greater than or equal to 2, $\Aut (X_0(D, N))$ is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions $W_0(D, N)$ as a subgroup of index 1 or 2. It is conjectured that $\Aut (X_0(D, N)) = W_0(D, N)$ except for finitely many values of $(D, N)$ and we provide criteria that allow us to show that this is indeed often the case. Our met…

On the cohomology of certain non-compact Shimura varieties (with an appendix by Robert Kottwitz)

The goal of this paper is to calculate the trace of the composition of a Hecke correspondence and a (high enough) power of the Frobenius at a good place on the intersection cohomology of the Satake-Baily-Borel compactification of certain Shimura varieties, to stabilize the result for Shimura varieties associated to unitary groups over $\mathbb{Q}$ and to give applications of this calculations using base change from these unitary groups to $GL_n$. ----- Le but de ce texte est de calculer la trace d'une correspondance de Hecke composee avec une puissance (assez grande) du Frobenius en une bonne …

关于非紧Unitary Shimura簇上同调的一般部分

On the generic part of the cohomology of non-compact unitary Shimura varieties

本文证明了偶数维quasi-split unitary群相关的非紧Shimura簇的模l上同调一般部分集中在中度以上，并通过计算紧化Shimura簇上Hodge-Tate周期映射的纤维，推导出无自对偶假设的potential automorphy定理。

Valeurs multiples de fonctions L de formes modulaires

This doctoral thesis studies the overlap between two well-known collections of results in number theory: the theory of periods and period polynomials of modular forms as developed by Eichler, Shimura and Manin and its extensions by Köhnen and Zagier, and the theory of 'multiple zeta values' (MZV's) as initied by Euler and studied by many authors in the last two decades. These two theories had been linked by Manin, who introduced 'multiple L-values' (MLV's). We propose to study those families of number and especially the relations among them, generalizing the Eichler-Shimura-Manin relations, an…

On the cohomology of compact unitary group Shimura varieties at ramified split places

In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in previous work, following the stabilization…

Shimura簇的缓增电流与Deligne上同调，及其在$\mathrm{GSp}_6$上的应用

Tempered currents and Deligne cohomology of Shimura varieties, with an application to $\mathrm{GSp}_6$

本文利用缓增电流给出任意Shimura簇Deligne-Beilinson上同调的新描述，并应用于计算Siegel六维簇motivic上同调类的Beilinson高阶调节子像，将其与Spin $L$-函数非临界特殊值联系，验证Beilinson猜想。

Vanishing theorems for the mod $p$ cohomology of some simple Shimura varieties

We show that the mod $p$ cohomology of a simple Shimura variety treated in Harris-Taylor's book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree.

An isomorphism between polynomial eigenfunctions of the transfer operator and the Eichler cohomology for modular groups

For the group PSL(2,Z) it is known that there is an isomorphism between polynomial eigenfunctions of the transfer operator for the geodesic flow and the Eichler cohomology in the theory of modular forms. In a recent paper by Chang and Mayer it is indicated that such an isomorphism exists as well for certain subgroups of PSL(2,Z). We prove this and provide some evidence by computer aided algebraic calculations that such an isomorphism exists for all principal congruence subgroups and all congruence subgroups of Hecke type.

On subquotients of the etale cohomology of Shimura varieties

We study the conditions imposed by conjectures of Arthur and Kottwitz on the Galois representations occurring in the cohomology of Shimura varieties.

On the generic part of the cohomology of compact unitary Shimura varieties

The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties.

Hodge型Shimura簇的局部解析完备上同调

Locally analytic completed cohomology of Shimura varieties of Hodge type

本文证明了Hodge型Shimura簇的局部解析完备上同调群与旗簇上同调群间的自然同构，推广了模曲线与酉Shimura曲线的相关结论。

On the $l$-adic cohomology of some $p$-adically uniformized Shimura varieties

We determine the Galois representations inside the $l$-adic cohomology of some unitary Shimura varieties at split places where they admit uniformization by finite products of Drinfeld upper half spaces. Our main results confirm Langlands-Kottwitz's description of the cohomology of Shimura varieties in new cases.

Igusa 堆栈与 Shimura 变分的上同调 II

Igusa Stacks and the Cohomology of Shimura Varieties II

本文构建了所有abelian型Shimura变分的Igusa堆栈，并推导了这些变分上同调的结论。作者证明了Fargues-Scholze局部Langlands对应与Arthur、Mok等构造的半简化局部Langlands对应在经典群A、B和D类型上一致。

Weight zero Eisenstein cohomology of Shimura varieties via Berkovich spaces

This brief article gives an alternative interpretation, based on a theorem of Berkovich, of the Eisenstein classes in the cohomology of Shimura varieties, used in forthcoming work of the author with K. W. Lan, R. Taylor, and J. Thorne.

Shimura曲线上的闭测地线交数计数

Counting intersection numbers of closed geodesics on Shimura curves

本文研究Shimura曲线上的闭测地线交数计数，通过实二次类比方法，推导出总交数公式，并提供非最大Eichler秩序和非基本判别式的通用结果。

Shimura 变种的有理完成上同调的一些消失结果

Some vanishing results for the rational completed cohomology of Shimura varieties

基于Bhatt在混合特征下的几乎 Kodaira 型消失结果，研究显示Shimura变种的局部解析完成上同调中，足够规则的微分权只能出现在中间度数。

Automorphic congruences and torsion in the cohomology of a simple unitary Shimura variety

We first give a relative flexible process to construct torsion cohomology classes for Shimura varieties of Kottwitz-Harris-Taylor type with coefficient in a non too regular local system. We then prove that associated to each torsion cohomology class, there exists a infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent.

Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations

We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some "generalized Hasse invariants" adapted to the Ekedahl-Oort stratification of the special fiber.

关于阿贝尔型 Shimura 簇上同调的泛部分

On the generic part of the cohomology of Shimura varieties of abelian type

本文研究了阿贝尔型 Shimura 簇上同调的泛部分，证明了挠消失结果，证实了 Hamann--Lee 猜想。其证明利用了 unipotent categorical local Langlands correspondence，且不依赖于表示的自同构分类或迹公式技术。

Motivic cohomology of quaternionic Shimura varieties and level raising

We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level raising between Hilbert modular forms. The main ingredient for our construction is a form of Ihara's Lemma for compact quaternionic Shimura surfaces which we prove by generalizing a method of Diamond-Taylor. Along the way we also verify the Hecke orbit conjecture for these quaternionic Shimura varieties which is a key input for our proof of Ihara's Lemma.

Algebraic de Rham theory for weakly holomorphic modular forms of level one

We establish an Eichler-Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasi-periods of modular forms.

Hecke operators and the coherent cohomology of Shimura varieties

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain $p$-adic estimates of Satake parameters of certain non-regular self dual automorphic representations of $\mathrm{GL}_n$.

Torsion classes in the cohomology of KHT Shimura varieties

A particular case of Bergeron-Venkatesh's conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of Shimura varieties, we first associate to each such torsion class an infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent. Then, using completed cohomology, we construct torsion classes in regular weight and then deduce explicit examples of such automorphic congruences.

On the cohomology of some simple Shimura varieties with bad reduction

We determine the Galois representations inside the $l$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.

Cohomology vanishing for Ekedahl--Oort strata on Hilbert--Blumenthal Shimura varieties

We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl--Oort strata of the special fiber of Hilbert--Blumenthal Shimura varieties. This work vastly extends to all Ekedahl--Oort strata previous results of Diamond--Kassaei and Goldring--Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\leq 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring--Koskivirta on the set of possible weights of automorphic forms.

De Rham $F$-gauge 与 Shimura 簇

De Rham $F$-gauges and Shimura varieties

本文研究特征p下光滑代数簇的de Rham上同调的F-gauge，构造带G结构的泛de Rham F-gauge以恢复Hodge型Shimura簇的Ekedahl-Oort分层，并在PEL型情形下通过相伴对偶BGG复形确定了其de Rham上同调的F-gauge结构。

p-adic Hodge-theoretic properties of étale cohomology with mod p coefficients, and the cohomology of Shimura varieties

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.

关于特征标为(1,n)的紧单位元Shimura簇上同调的非通用部分

On the non-generic part of cohomology of compact unitary Shimura varieties of signature $(1,n)$

本文研究特征标为(1,n)的紧单位元Shimura簇的非通用部分同调，扩展Boyer的结果，采用Fargues-Scholze的工作和Koshikawa的方法。

Galois irreducibility implies cohomology freeness for KHT Shimura varieties

Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{ρ_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2iπ/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimur…

The stable trace formula for Shimura varieties of abelian type

We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.

A remark on a conjecture of Buzzard-Gee and the cohomology of Shimura varieties

We compare the conjecture of Buzzard-Gee on the association of Galois representations to C-algebraic automorphic representations with the conjectural description of the cohomology of Shimura varieties due to Kottwitz, and the reciprocity law at infinity due to Arthur. This is done by extending Langlands's representation of the L-group associated with a Shimura datum to a representation of the C-group of Buzzard-Gee. The approach offers an explanation of the explicit Tate twist appearing in Kottwitz's description.