We prove the compatibility of local and global Langlands correspondences for GL_n, which was proved up to semisimplification by Harris-Taylor. More precisely, for the n-dimensional l-adic representation R_l(Π) of the Galois group of a CM-field L attached to a conjugate self-dual regular algebraic cuspidal automorphic representation Π, which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of R_l(Π) to the decomposition group of a prime v of L not dividing l corresponds to Π_v by the local Langlands correspondence. As a corollary, we show the irreducibility of R_l(Π) when Π_v is square integrable for some finite place v outside l. We also obtain conditional results in the case v|l.
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.
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.
After observing that some constructions and results in the $p$-adic Langlands programme are somehow independent from $p$, we formulate the hypothesis that this astonishing uniformity could be explained by a 1-adic Langlands correspondence.
We study torsion in the homology of arithmetic groups and give evidence that it plays a role in the Langlands program. We prove, among other results, a numerical form of a Jacquet--Langlands correspondence in the torsion setting.
We consider a novel version of the classical Jacquet-Langlands {correspondence}, explore a $p$-adic extension of the Jacquet-Langlands correspondence, and as an explicit example we find an overconvergent automorphic form of weight~1 which corresponds to a classical modular form of weight~1, using both experimental and theoretical methods.
We use the patching method of Taylor--Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL_n(F), F a finite extension of Q_p. We use our construction to prove many new cases of the Breuil--Schneider conjecture.
We extend the results by R.P. Langlands on representations of (connected) abelian algebraic groups. This is done by considering characters into any divisible abelian topological group. With this we can then prove what is known as the abelian case of the $p$-adic Langlands program.
We study the variation of the local Langlands correspondence for ${\rm GL}_{n}$ in characteristic-zero families. We establish an existence and uniqueness theorem for a correspondence in families, as well as a recognition theorem for when a given pair of Galois- and reductive-group- representations can be identified as local Langlands correspondents. The results, which can be used to study local-global compatibility questions along eigenvarieties, are largely analogous to those of Emerton, Helm, and Moss on the local Langlands correspondence over local rings of mixed characteristic. We apply th…
We use results by Chenevier to interpolate the classical Jacquet-Langlands correspondence for Hilbert modular forms, which gives us an extension of Chenevier's results to totally real fields. From this we obtain an isomorphisms between eigenvarieties attached Hilbert modular forms and those attached to modular forms on a totally definite quaternion algebra over a totally real field of even degree.
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).
We prove that the reduction mod \ell of the local Langlands correspondence between supercuspidal representations of GL_n(F), where F is a finite extension of Q_p, and representations of the Galois group of F is well-defined. The results and methods of this paper have been used by Vigneras to give a proof of the ``local Langlands conjecture mod \ell'', that was proved earlier by her by different methods (local harmonic analysis mod \ell) for \ell>n. Unlike the local methods used earlier by Vigneras, our more global method also generalises well to the characteristic p case.
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.
We prove a descent criterion for certain families of smooth representations of GL_n(F) (F a p-adic field) in terms of the gamma factors of pairs constructed in previous work of the second author. We then use this descent criterion, together with a theory of gamma factors for families of representations of the Weil group W_F (developed previously by both authors), to prove a series of conjectures, due to the first author, that give a complete description of the integral Bernstein center in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural…
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.
Let E be a nonarchimedean local field with residue characteristic l, and suppose we have an n-dimensional representation of the absolute Galois group G_E of E over a reduced complete Noetherian local ring A with finite residue field k of characteristic p different from l. We consider the problem of associating to any such representation an admissible A[GL_n(E)]-module in a manner compatible with the local Langlands correspondence at characteristic zero points of Spec A. In particular we give a set of conditions that uniquely characterise such an A[GL_n(E)]-module if it exists, and show that su…
Using a level-raising argument (and a result of Larsen on the image of Galois representations in compatible systems), we prove that for any automorphic representation $π$ for $\U(3)$, the $l$-adic Galois representation $ρ_l$ which is attached to $π$ by the work of Blasius and Rogawski, is the one expected by local Langlands correspondance at every finite place (at least up to semi-simplification and for a density one set of primes $l$). We rely on the work of Harris and Taylor, who have proved the same results (for $\U(n)$) assuming the base change of $π$ is square-integrable at one place. As …
We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and L-values of elliptic modular forms for prime level in weight 2 both to nonsquare level and to Hilbert modular forms.
Let $F$ be a $p$-adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell}$, with $\ell$ different from $p$. We define ``nilpotent lifts'' of irreducible generic $k$-representations of $GL_n(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $π$ of $GL_n(F)$ is uniquely determined by its collection of Rankin--Selberg gamma factors $γ(π\times \widetildeτ,X,ψ)$ as $\widetildeτ$ varies over nilpotent lifts of irreducible generic $k$-representations $τ$ of $GL_t(F)$ for $t=1,\dots, \lfloor \frac{n}{2}\rfloor$. This…
The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for GL(n) over a nonarchimedean local field. In this article we make progress towards a purely local proof of this fact. To wit, we find a family of open affinoid subsets of Lubin-Tate space at infinite level, whose cohomology realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree n extension). A key role is played by a certain variety X, defined over a finite field, which is "maximal" in the sense that the num…
We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of GL(n) over a non-archimedean local field by means of the local Jacquet-Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.
We generalize the rectifier of Bushnell and Henniart, which occurs in the local Langlands correspondence for $GL_n(K)$, to certain Langlands parameters for unramified connected reductive groups.
The goal of this paper is to reformulate the conjectural "Ihara lemma" for $U(n)$ in terms of the local Langlands correspondence in families $\tildeπ_Σ(\cdot)$, as currently being developed by Emerton and Helm. The reformulation roughly takes the following form. Suppose we are given an irreducible mod $\ell$ Galois representation $\bar{r}$, which is modular of full level (and small weight), and a finite set of places $Σ$ -- none of which divide $\ell$. Then $\tildeπ_Σ(r)$ exists, and has a global realization as a natural module of algebraic modular forms, where $r$ is the universal $Σ$-deforma…
We prove the local Langlands correspondence for GSp_4(F), where F is a non-archimedean local field of positive characteristic with residue characteristic > 2.
In this paper, we continue the work of the first author and give a new construction of the tame local Langlands correspondence for PGSp(4,F), where F is a p-adic field, that is analogous to the construction of the local Langlands correspondence for real groups.
In this paper, we give a method for characterizing the local Langlands conjectures in the vein of Scholze's alternate proof of the local Langlands conjecture for $\mathrm{GL}_n$. More specifically, we show that if a local Langlands correspondence satisfies a Scholze--Shin equation, as in the paper of Scholze and Shin, in addition to the usual desiderata expected of such a correspondence then these properties uniquely characterize the correspondence.
Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover,we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe-Gan for quasi-split even orthogonal groups.
We give an explicit description of the modified mod p local Langlands correspondence for GL_2(Q_{\ell}) of Emerton-Helm, where p is an odd prime different from \ell.
In the paper as a new application of the Jacquet-Langlands correspondence we connect the transfer operators for different cofinite Fuchsian groups by comparing the corresponding Selberg zeta functions.
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
We prove that Kaletha's local Langlands correspondence for regular supercuspidal representations gives the classical local Jacquet--Langlands correspondence due to Deligne--Kazhdan--Vigneras and Badulescu. As in a former joint paper with Oi, where a similar result is proved for the local Langlands correspondence for the general linear group, the key ingredients in our proof are the work of Bushnell--Henniart explicitly describing the local Jacquet--Langlands correspondence for essentially tame supercuspidal representations and its reinterpretation due to Tam in terms of Langlands--Shelstad's \…
From $p$-modular to $p$-adic Langlands correspondences for $\operatorname{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$: deformations in the non-supercuspidal case
In this paper, we show the Langlands correspondence for isocrystals on curves. This shows the existence of crystalline companion in the curve case. For the proof, we construct the theory of arithmetic $\mathscr{D}$-modules for algebraic stacks whose diagonal morphisms are finite.
We prove a result which provides a link between the decomposition of parabolically induced representations and the Bushnell--Kutzko theory of typical representations. As an application, we show that there exists a well-defined inertial Langlands correspondence which respects the monodromy action of L-parameters, under some standard conjectures regarding the local Langlands correspondence. To allow for potential applications of this inertial Langlands correspondence, we also provide a complete construction of the set of typical representations, giving a parametrization of these in terms of the …
In this paper, we extend our result on a depth preserving property of the local Langlands correspondence for quasi-split unitary groups (arXiv:1804.10901) to non-quasi-split unitary groups by using the local theta correspondence. The key ingredients are a depth preserving property of the local theta correspondence proved by Pan and a description of the local theta correspondence via the local Langlands correspondence established by Gan--Ichino. To combine them, we compare splittings for metaplectic covers of unitary groups constructed by Kudla with those constructed by Pan.
In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).
In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.
We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet-Langlands correspondence. Along the way, we prove a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes.
Let $F$ be a a non-Archimedean local field of characteristic 0 and $G$ be an inner form of the general linear group $G^*=\mathrm{GL}_n$ over $F$. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for $G$ and $G^*$ can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langlands and Shelstad. As a consequence, the essentially tame local Langlands correspondence for $G$ can be described using admissible embeddings of L-tori.
In this paper, we prove the coincidence of Kaletha's recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris--Taylor's one in the case of general linear groups. The keys are Bushnell--Henniart's essentially tame local Langlands correspondence and Tam's result on Bushnell--Henniart's rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha's and Tam's $χ$-data.
We describe two candidates for a local p-adic Jacquet-Langlands correspondence and using patching we show that they are in fact isomorphic. We then study locally algebraic vectors of the given correspondence.
In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O_{2n} established by Arthur. We give two applications: Prasad's conjecture and Gross--Prasad conjecture for O_{n}. Also, we discuss the Arthur conjecture for O_{2n}, and establish the Arthur multiplicity formula for O_{2n}.
These are expanded notes of some lectures given by the author for a workshop held at the Indian Statistical Institute, Bangalore in June, 2010, giving an exposition on the modular representations of finite groups of Lie type and $p$-adic groups, and the modular Langlands correspondence. These notes give an overview of the subject with several examples.
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with $p\neq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of the general linear group $\text{\rm GL}_n(F)$. We give a transparent parametrization of the irreducible, totally ramified, cuspidal representations of $G$ of parametric degree $n$. We show that the parametrization is respected by the Jacquet-Langlands correspondence, relative to any other inner form. This expresses the Jacquet-Langlands correspondence for such representations within a single, compact formula.
Following Weinstein, Boyarchenko-Weinstein and Imai-Tsushima, we construct a family of affinoids in the Lubin-Tate perfectoid space and formal models such that the cohomology of the reduction of each formal model realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for certain representations. In the terminology of the essentially tame local Langlands correspondence, the representations treated here are characterized as being parametrized by minimal admissible pairs in which the field extensions are totally ramified.
We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. The novelty is we allow non-classical points, possibly non-étale over weight space. More precisely we interpolate the local Langlands correspondence for GL(n) across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Scholze from his new approach to the local Langlands conjecture.
