Thomas Barnet-Lamb, Toby Gee, David Geraghty, Richard Taylor
We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GL_n over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight.
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We prove some new cases of local--global compatibility for the Galois representations associated to Hilbert modular forms of low weight (that is, partial weight one).
We prove the compatibility at places dividing l of the local and global Langlands correspondences for the l-adic Galois representations associated to regular algebraic essentially (conjugate) self-dual cuspidal automorphic representations of GL_n over an imaginary CM or totally real field. We prove this compatibility up to semisimplification in all cases, and up to Frobenius semisimplification in the case of Shin-regular weight.
We generalize the local-global compatibility result in arXiv:1506.04022 to higher dimensional cases, by examining the relation between Scholze's functor and cohomology of Kottwitz-Harris-Taylor type Shimura varieties. Along the way we prove a cuspidality criterion from type theory. We also deal with compatibility for torsion classes in the case of semisimple mod $p$ Galois representations with distinct irreducible components under certain flatness hypotheses.
Let $F_{\wp}$ be a finite extension of $\mathbb{Q}_p$. By considering partially de Rham families, we establish a Colmez-Greenberg-Stevens formula (on Fontaine-Mazur $\mathcal{L}$-invariants) for (general) $2$-dimensional semi-stable non-crystalline $\mathrm{Gal}(\overline{\mathbb{Q}_p}/F_{\wp})$-representations. As an application, we prove local-global compatibility results for completed cohomology of quaternion Shimura curves, and in particular the equality of Fontaine-Mazur $\mathcal{L}$-invariants and Breuil's $\mathcal{L}$-invariants, in critical case.
We prove that certain Galois-isotypic parts of the completed cohomology group for U(2) can be written as a completed tensor product of a representation coming from the p-adic Langlands correspondence for $GL_2(\mathbb{Q}_p)$ and a representation arising via the local Langlands correspondence in famillies of Emerton and Helm.
An Ordinary Rank-Two Case of Local-Global Compatibility for Automorphic Representations of Arbitrary Weight Over CM Fields
本文证明了满足普通条件的模 l 表示的秩二潜在自守性定理，并结合普通自守提升定理，确立了复数域上任意权正则代数尖点自守表示在 p≠l 情形下的局部-整体相容性。
A Rank-Two Case of Local-Global Compatibility for $l = p$
本文针对 CM 域上 GL$_2$ 的权 0 正则代数尖点自守表示，利用自守性提升定理证明 $l=p$ 局部-整体相容性，并据此定义 Fontaine–Mazur $\mathcal{L}$-不变量。
The completed Kirillov model and local-global compatibility for functions on Igusa varieties
本文将GL2的普通Igusa簇上的尖点函数描述为经典模形式Kirillov模型的完备化，建立弱局部-全局相容性定理，并猜想更一般情形下的Hida理论类似物。
Local-global compatibility of automorphic Galois representations over CM fields at $p$
研究 CM 数域上\(\mathrm{GL}_n(\mathbf{A}_F)\)自守表示相关的\(p\)进伽罗瓦表示，证明其在\(p\)进霍奇理论下潜在半稳定且与局部朗兰兹对应在半简化意义下相容。
Compatibility of local and global Langlands correspondences
本文证明了GL_n的局部和整体Langlands对应的相容性，Harris-Taylor半简化地证明了这一点。更精确地说，对于CM域L的伽罗瓦群的n维l-adic表示R_l（R_l），它与共轭自对偶正则代数尖形自守表示R_l（R_l）相连，R_l（R_l）在有限处平方可积〔的〕，我们证明了将R_l（R_l）限制到L的素数v的不整除l的分解群的Frobenius半简化对应于局部Langlands对应.作为推论，我们证明了当n_v对l外的某个有限位置v是平方可积〔的〕时，R_l（n）的不可约性.我们也得到了条件结果的情况下v| L.
We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(π)$ denote an $n$-dimensional $p$-adic representation of the Galois group of a CM field $F$ attached to a regular algebraic cuspidal automorphic representation $π$ of $GL_n(\mathbb{A}_F)$. We show that the restriction of $r_p(π)$ to the decomposition group of a place $v\nmid p$ of $F$ corresponds up to semisimplification to $rec(π_v)$, the image of $π_v$ under the local Langlands…
Local to Global Compatibility on the Eigencurve (l not equal p)
推广Coleman的Hecke算子构造，定义了GL_2（Q_l）在有限斜率过收敛p-adic模形式（l不等于p）空间上的作用.这样，我们把GL_2（Q_1）的一个可容许光滑表示，推广了经典的构造，与Tame水平N的Coleman-Mazur特征曲线上的任何C_p值点相联系。使用伽罗瓦理论解释的特征曲线，我们关联的2维威尔-德利涅表示这样的点，并表明，远离一个离散集，他们同意根据当地朗兰兹对应。
Ordinary parts and local-global compatibility at $\ell=p$
本文通过建立任意抛物子群下 $Q$-普通自对偶自守表示的相容性，证明并推广 Scholze 挠自守 Galois 表示在 $\ell=p$ 处的局部-整体相容性。
Let $F$ be a totally real number field, $\wp$ a place of $F$ above $p$. Let $ρ$ be a $2$-dimensional $p$-adic representation of $\mathrm{Gal}(\bar{F}/F)$ which appears in the étale cohomology of quaternion Shimura curves (thus $ρ$ is associated to Hilbert eigenforms). When the restriction $ρ_{\wp}:=ρ|_{D_{\wp}}$ at the decomposition group of $\wp$ is semi-stable non-crystalline, one can associate to $ρ_{\wp}$ the so-called Fontaine-Mazur $\mathcal{L}$-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these $\mathcal{L}…
We strengthen the compatibility between local and global Langlands correspondences for GL_{n} when n is even and l=p. Let L be a CM field and Π a cuspidal automorphic representation of GL_{n}(\mathbb{A}_{L}) which is conjugate self-dual and regular algebraic. In this case, there is an l-adic Galois representation associated to Π, which is known to be compatible with local Langlands in almost all cases when l=p by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless Π has Shin-regular weight. We extend the compatibility to Frobe…
We remove the semisimple condition in the mod-$p$ local-global compatibility result of arXiv:2106.10674. Namely, assuming flatness of $π_{\mathfrak{m}}^\vee$ and $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ being multiplicity free, we prove that $H^{n-1}_{\text{ét}}(\mathbb{P}_{\mathbb{C}_p}^{n-1}, \mathcal{F}_{π_{\mathfrak{m}}[\mathfrak{m}]})$ determines $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ uniquely. We give remarks on our assumptions at the end of this note.
We introduce a general formalism with minimal requirements under which we are able to prove the pro-modular Fontaine-Mazur conjecture. We verify it in the ordinary case using the recent construction of Breuil and Herzig.
On the potential automorphy and the local-global compatibility for the monodromy operators at $p \neq l$ over CM fields
本文研究了CM域上非l素数处GL_n(A_F)的自守表示的局部-整体兼容性及高维情况下的拉马努金猜想证明。
Inspired by Emerton's work for GL(2), we study the completed cohomology of the tower of finite sets associated with a definite unitary group in two variables. When p splits (and other technical assumptions are fulfilled), we show that the p-adic local Langlands correspondence for GL(2) (over Q_p) occurs in the cohomology. We give an application to the Fontaine-Mazur conjecture over CM fields.
Local-global compatibility at $p\neq\ell$ for torsion automorphic forms
本文推广 Varma 的结果至 Betti 上同调中的挠类，结合 Scholze 构造与 $\mathbf{Z}_{\ell}$ 系数 $p$ 进一般线性群表示论，证明了 $p\neq\ell$ 处自守群行列式的局部-整体兼容性。
Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois representation such that rbar|_{Gal(F_w-bar/F_w)} is upper-triangular, maximally non-split, and generic. If rbar is automorphic, and some suitable technical conditions hold, we show that rbar|_{\Gal(F_w-bar/F_w)} can be recovered from the GL_3(F_w)-action on a space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights i…
We strengthen the local-global compatibility of Langlands correspondences for $GL_{n}$ in the case when $n$ is even and $l\not=p$. Let $L$ be a CM field and $Π$ be a cuspidal automorphic representation of $GL_{n}(\mathbb{A}_{L})$ which is conjugate self-dual. Assume that $Π_{\infty}$ is cohomological and not "slightly regular", as defined by Shin. In this case, Chenevier and Harris constructed an $l$-adic Galois representation $R_{l}(Π)$ and proved the local-global compatibility up to semisimplification at primes $v$ not dividing $l$. We extend this compatibility by showing that the Frobenius …
Sur la compatibilite entre les correspondances de Langlands locale et globale pour U(3). (On the compatibility between local and global Langlands correspondances for U(3))
用一个提高等级的论点（和Larsen关于相容系统中伽罗瓦表示的图像的结果），我们证明了对于$\U(3)$的任何自守表示$π$，由Blasius和Rogawski的工作附加到$π$的$l$ -adic伽罗瓦表示$ρ_l$，是一个预期的本地朗兰兹对应在每一个有限的地方（至少到半简化和密度一组素数$l$）。我们依赖于Harris和Taylor的工作，他们已经证明了相同的结果（对于$\U(n)$），假设$π$的基变化在一个地方是平方可积的。作为一个推论，每一个在无穷多个位置上被调和的自守表示在每一个位置上都被调和。
Local-global compatibility and the exceptional zero conjecture for GL(3)
本文研究 GL(3) 中 p 处 Steinberg 的 p-常自守表示，通过算术上同调证明自守例外零点猜想，并借助 GL_n 挠类在 p 处的局部-整体相容性推出 L-不变量相等。
Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation $\overline{r}:\mathrm{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ is upper-triangular and satisfies certain genericity conditions at a place $w$ above $p$, and that every subquotient of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ of dimension $>2$ is Fontaine--Laffaille generic. In this paper, we show that the isomorphism class of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ …
Let $ρ_p$ be a $3$-dimensional $p$-adic semi-stable representation of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2\ne 0$ on $D_{\mathrm{st}}(ρ_p)$. When $ρ_p$ comes from an automorphic representation $π$ of $G(\mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $\mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(\mathbb{A}_{F^+}^\…
Local-global compatibility over function fields
本文证明 V. Lafforgue 整体 Langlands 对应与 Fargues-Scholze 半简化局部 Langlands 对应兼容，并借此将后者典范提升为非半简化对应，同时证明其与 Genestier-Lafforgue 构造一致。
Compatibility of canonical $\ell$-adic local systems on Shimura varieties, II
本文将先前关于Shimura簇上典范局部系统在伴随群投影下相容性的结果，加强为证明$G(\mathbb{Q}_{\ell})$-局部系统本身的相容性，并进一步包含了crystalline相容性。
Mod $p$ local-global compatibility for $\mathrm{GSp}_4(\mathbb{Q}_p)$ in the ordinary case
本文在偶次全实域与常情形、一般性及 Taylor–Wiles 条件下，证明模 $p$ 的 $\mathrm{GSp}_4$ 自守形式 Hecke 同型部分上的局部群作用可决定相应 Galois 表示在 $p$ 位的限制。
Moduli of Fontaine--Laffaille representations and a mod-$p$ local-global compatibility result
本文在 Taylor–Wiles 条件下证明模 p 局部-整体相容性：无限层模 p 上同调中的 GL_n(F_{\widetilde{v}}) 作用可决定 Fontaine–Laffaille 且半简化一般的局部 Galois 表示。
Compatibility of $F$-isocrystals on adjoint Shimura varieties
研究伴随志村簇上典范\(\ell -\)进局部系和典范\(F -\)等晶体的相容性，方法依赖典范\(p -\)进局部系晶体性、Margulis超刚性及相关构造，扩展了前两位作者过去的结果。
Multivariable ($φ$,$\mathcal{O}_K^\times$)-modules and local-global compatibility
该研究利用完美胚空间，将有限维伽罗瓦连续表示对应到特定完备局部化环上的平展(φ, O_K^×)-模，提出志村曲线模p上同调中GL₂(K)光滑表示的对应猜想，并在半单且足够一般情形下予以证明。
Kudla-Rapoport conjecture for unramified maximal parahoric level
本文通过局部-全局相容性将Kudla-Rapoport猜想化为全局相交问题，并利用特殊除子生成级数的模性进行归纳，证明了无分歧酉群极大抛物层级的该猜想。
Compatibility of the Fargues--Scholze correspondence for unitary groups
本文证明了酉群的 Fargues--Scholze 对应与 Mok 的局部 Langlands 对应相容，并结合谱作用验证了范畴形式猜想，进而推出 Fargues 特征层猜想和最强 Kottwitz 猜想。
Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands
本文证明了在非分歧且p>2的条件下，Fargues-Scholze构造的半单Weil参数与Gan-Takeda关于GSp4及其非分裂内形式的局部Langlands对应相兼容，并利用局部Shimura簇上同调验证了Kottwitz猜想的强形式。
Compatibility of canonical $\ell$-adic local systems on Shimura varieties
本文针对超刚性情形的志村簇与整洁水平子群，证明了数域点对应的典范ℓ进表示族构成相容系，该结果由连通志村簇上典范局部系统的更强相容性推导而来，首次给出非阿贝尔型志村簇的ℓ独立性结论。
We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility properties of the refined local Langlands correspondence.
We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations of Gal_F x D^*, where Gal_F is the absolute Galois group of F, and D/F is the central division algebra of invariant 1/n. Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.
We prove that the conjectural depth zero local Langlands correspondence of DeBacker/Reeder agrees with the depth zero local Langlands correspondence as described by Moy, for the group $GL(\ell,F)$, where $\ell$ is prime and F is a local non-archimedean field of characteristic 0. We also prove that if one assumes a certain compatibility condition between Adler's and Howe's construction of supercuspidal representations, then the conjectural positive depth local Langlands correspondence of Reeder also agrees with the positive depth local Langlands correspondence as described by Moy, for $GL(\ell,…
We give a proof of the Breuil-Schneider conjecture in a large number of cases, which complement the indecomposable case, which we dealt with earlier in [Sor]. In some sense, only the Steinberg representation lies at the intersection of the two approaches. In this paper, we view the conjecture from a broader global perspective. If $U_{/F}$ is any definite unitary group, which is an inner form of $\GL(n)$ over $\K$, we point out how the eigenvariety $\X(K^p)$ parametrizes a global $p$-adic Langlands correspondence between certain $n$-dimensional $p$-adic semisimple representations $ρ$ of $\Gal(\…
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