On Langlands functoriality- reduction to the semistable case

First steps towards $p$-adic Langlands functoriality

By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the …

Langlands functoriality conjecture for ${\bf SO}_{2n}^*$ in positive characteristic

In this article, we are concerned with the Langlands functoriality conjecture. Cogdell, Kim, Piatetski-Shapiro and Shahidi proved functioriality conjecture in the case of a globally generic cuspidal automorphic representation for the split classical groups, unitary groups or even quasi-split special orthogonal groups in characteristic zero. Lomelí extends this result to split classical groups and unitary groups in positive characteristic. Thus, in this article we prove the Langlands functoriality conjecture for the even quasi-split non-split special orthogonal groups in positive characteristic…

Langlands Program, Trace Formulas, and their Geometrization

The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey of my recent joint work with Robert Langlands and Ngo Bao Chau (arXiv:1003.4578 and arXiv:1004.5323) on a new approach to proving the Functoriality Conjecture using the trace formulas, and on the geometr…

Universal eigenvarieties, trianguline Galois representations, and p-adic Langlands functoriality

Using the overconvergent cohomology modules introduced by Ash and Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash-Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of \emph{p-}adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequen…

Irreducible components of extended eigenvarieties and interpolating Langlands functoriality

We study the basic geometry of a class of analytic adic spaces that arise in the study of the extended (or adic) eigenvarieties constructed by Andreatta--Iovita--Pilloni, Gulotta and the authors. We apply this to prove a general interpolation theorem for Langlands functoriality, which works for extended eigenvarieties and improves upon existing results in characteristic 0. As an application, we show that the characteristic p locus of the extended eigenvariety for GL(2)/F, where F is a cyclic extension of the rational numbers Q, contains non-ordinary components of dimension at least [F:Q].

Automorphic L-functions and functoriality

This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising cases of Langlands Functoriality Conjecture over number fields. They have led to striking new estimates towards Ramanujan and Selberg conjectures.

Spherical varieties, functoriality, and quantization

We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical $G$-variety $X$, one associates a dual group ${^LG_X}$ and an $L$-value (encoded in a representation of ${^LG_X}$), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism ${^LG_X}\to {^LG_Y}$ of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approac…

Functoriality and special values of L-functions

This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of $L$-functions. The emphasis is on symmetric power $L$-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic $L$-functions.

一个弗兰克滤波器的 functorial 细化及其对于自守形式空间的雅各布-朗格兰斯对应关系

A Functorial Refinement of the Franke Filtration and the Jacquet--Langlands Correspondence for Spaces of Automorphic Forms

本文通过扩展弗兰克滤波器的functorial细化，建立了自守形式空间中一般线性群及其内形式之间的雅各布-朗格兰斯对应关系，扩展了Badulescu的对应关系并包含完整functorial提升。

Relative functoriality and functional equations via trace formulas

Langlands' functoriality principle predicts deep relations between the local and automorphic spectra of different reductive groups. This has been generalized by the relative Langlands program to include spherical varieties, among which reductive groups are special cases. In the philosophy of Langlands' "beyond endoscopy" program, these relations should be expressed as comparisons between different trace formulas, with the insertion of appropriate $L$-functions. The insertion of $L$-functions calls for one more goal to be achieved: the study of their functional equations via trace formulas. The…

Endoscopy and the automorphic tensor product on the unitary group

We study some tempered endoscopic cases of Langlands functoriality on the $n$-variable unitary groups via the simple stable trace formula. This extends previous work of Rogawski and Clozel-Harris-Labesse. Ramakrishnan and Kim-Shahidi have proved the existence of the automorphic tensor product for $GL_2 \times GL_2$ and $GL_2 \times GL_3$. We apply our endoscopic results to, in a sense, "lift" these theorems and deduce the existence of the automorphic tensor product for the quasi-split unitary groups $U^*_2 \times U^*_2$ and $U^*_2 \times U^*_3$. We then formulate the notion of overconvergent L…

Completed cohomology of Shimura curves and a p-adic Jacquet-Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet-Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur's `level lowering' principle.

本质上一元的超 cuspidal 表示的局部朗兰兹对应（对于不连通的半单群）

The local Langlands correspondence of essentially unipotent supercuspidal representations for disconnected reductive groups

本文在刚性内形式框架下构造了本质上一元的超 cuspidal 表示的局部朗兰兹对应，并证明了某种泛函性和兼容性，该结果比 [Sol20] 中的类似结果更强，因为后者不考虑内扭的刚性化。同时，本文将这种对应推广到不连通的半单群。

Generic Transfer for General Spin Groups

We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp(4) to GL(4) follows from our result as a special case.

The Trace Formula and the Proof of the Global Jacquet-Langlands Correspondence

This is a survey of the proof of the global Jacquet-Langlands correspondence between GL_n and general inner forms, in characteristic zero. The proof is given after a long introduction recalling in some detail the objects and results involved. The paper is based on the notes of the lectures I gave at the doctoral school in CIRM 2016 (Jean-Morlet Chair) organized by Dipendra Prasad, Volker Heiermann and Fiona Murnaghan. It appeared in 2018 in the volume Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms (LNM 2221).

On the images of the Galois representations attached to generic automorphic representations of GSp(4)

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for almost every prime. Moreover, by using the notion of (n,p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes.

Generic transfer from GSp(4) to GL(4)

We establish Langlands functoriality for the generic spectrum of GSp(4) and describe its transfer on GL(4). We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).

Local Base Change via Tate Cohomology

In this paper we propose a new way to realize cyclic base change (a special case of Langlands functoriality) for prime degree extensions of characteristic zero local fields. Let $F / E$ be a prime degree $l$ extension of local fields of residue characteristic $p \neq l$. Let $π$ be an irreducible cuspidal $l$-adic representation of $\mathrm{GL}_n(E)$ and $ρ$ be an irreducible cuspidal $l$-adic representation of $\mathrm{GL}_n(F)$ which is Galois-invariant. Under some minor technical conditions on $π$ and $ρ$ (for instance, we assume that both are level zero) we prove that the $\bmod l$-reducti…

解析Langlands对应的一般框架

A general framework for the analytic Langlands correspondence

本文讨论任意局部域上解析Langlands对应的一般框架，专门针对阿基米德情形用 opers 描述 Hecke 算子谱，提出函子性原理类似物，并将其工具应用于 Gaudin 模型及其 q-形变。

Functorial products for $GL_2\times GL_3$ and the symmetric cube for $GL_2$

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms on $GL_6$, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on $GL_2$. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain $L$-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence, we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field a…

Local-global compatibility and the action of monodromy on nearby cycles

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 …

全局Langlands对应中具有大像的Galois表示

Galois representations with large image in the global Langlands correspondence

研究GL_n上代数自守表示关联的Galois表示残差表示像的大小，利用标准技术证明对全实域及奇素数n≤293时无穷多素数ℓ像最大，并在F=ℚ, n=5时证明大像猜想。

On the automorphy of 2-dimensional potentially semi-stable deformation rings of $G_{\mathbb{Q}_p}$

Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_p)$, we prove that the support of patched modules constructed by Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin meet every irreducible component of the potentially semistable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_p$ when $p > 2$, which is new in the case $p = 3$ and $\bar{r}$ a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction …