We survey several recent examples of derived structures emerging in connection with the Langlands correspondence. Cases studies include derived Galois deformation rings, derived Hecke algebras, derived Hitchin stacks, and derived special cycles. We also highlight some open problems that we expect to be important for future progress.
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.
We reprove the Local Langlands Correspondence for $\GL_n$ over $p$-adic fields as well as the existence of $\ell$-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor. In contrast to the proofs of the Local Langlands Correspondence given by Henniart and Harris-Taylor our proof completely by-passes the numerical Local Langlands Correspondence of Henniart. Instead, we make use of a previous result describing the inertia-invariant nearby c…
Let p and l be two distinct primes. We show how, under a certain congruence hypothesis, the mod l cohomology of the Lubin-Tate tower together with a certain Lefschetz operator provides a geometric interpretation of Vigneras' Langlands correspondence for a p-adic field with mod l coefficients.
In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that n is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic techn…
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.
We propose a geometric strategy of giving explicit description of the Langlands parameter of an irreducible supercuspidal representation of GL(n) over a non-archimedean local field. The key is to compare the cohomology of an affinoid in the Lubin-Tate space at infinite level and that of the reduction of its formal model. As examples, we treat the cases of depth 0 supercuspidal representations and simple supercuspidal representations.
In this expository paper we provide a geometric proof of the local Langlands Correspondence for the groups $\operatorname{GL}_{1}$ defined over $p$-adic fields $K$. We do this by redeveloping the theory of proalgebraic groups and use this to derive local class field theory in the style of Serre and Hazewinkel. In particular, we show that the local class field theory of Serre and Hazewinkel is valid for both equal characteristic and mixed characteristic ultrametric local fields. Finally, we use this to prove an equivalence of the categories of smooth representations of $K^{\ast}$ with continuou…
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.
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 \…
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.
We generalize a beautiful method of Blasius and Ramakrishnan, that in order to exhibit particular instances of the Langlands functorial correspondence, it is enough to show that the correspondence holds in the semistable case, provided the arithmetic data we are considering is closed with respect to cyclic base change and descent.
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.
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.
Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. We use an explicit description of the correspondence from an earlier paper, and otherwise entirely elementary methods, to show that it respects the relationship of congruence modulo $\ell$. The $\ell$-modular correspondence thereby becomes as effective as the complex one.
Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which was established in an early paper of Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this provides an alternative approach to the works of Kaletha-Minguez-Shin-White and Moeglin-Renard.
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 prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. The proof holds for tempered representations of quasi-split real K-groups and quasi-split p-adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.
Let $F$ be a non-archimedean local field. For any irreducible representation $π$ of an inner form $G'=\mathrm{GL}_{m}(D)$ of $G=\mathrm{GL}_{N}(F)$, there exists an irredubile representation of a maximal compact open subgroup in $G'$ which is also a type for $π$. Then we can consider the problem whether these types are unique or not in some sense. If such types for $π$ are unique, we say $π$ has the strong unicity property of types. On the other hand, there exists a correspondence connecting irreducible representations of $G'$ and $G$, called the Jacquet--Langland correspondence. In this paper…
Let $F$ be a locally compact non-Archimedean field, and let $B/F$ be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations of $B^\times$ of dimension $>1$ and irreducible cuspidal representations of $GL_2(F)$. We present a new construction of this bijection in which the preservation of epsilon factors is automatic.
We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic twisted Hilbert modular surface. This quantity is then related to the arithmetic volume of a Shimura curve, via the arithmetic Grothendieck-Riemann-Roch theorem and the Jacquet-Langlands correspondence.
Let F be a non-archimedean local field of characteristic zero with residual characteristic p. In this paper, we present a simple proof and construction of the local Langlands correspondence for simple supercuspidal representations of GL_n(F), when p does not divide n. As an application, we prove Jacquet's conjecture on the local converse problem for GL_n(F) in the case of simple supercuspidal representations, for arbitrary p.
In this paper we describe the unramified Langlands correspondence for two-dimensional local fields, we construct a categorical analogue of the unramified principal series representations and study its properties. The main tool for this description is the construction of a central extension. For this (and other) central extension we prove noncommutative reciprocity laws (i.e. the splitting of the central extensions over some subgroups) for arithmetic surfaces and projective surfaces over a finite field. These reciprocity laws connect central extensions which are constructed locally and globally…
The p-adic local Langlands correspondence for GL_2(Q_p) is given by an exact functor from unitary Banach representations of GL_2(Q_p) to representations of the absolute Galois group G_{Q_p} of Q_p. We prove, using characteristic 0 methods, that this correspondence induces a bijection between absolutely irreducible non-ordinary representations of GL_2(Q_p) and absolutely irreducible 2-dimensional representations of G_{Q_p}. This had already been proved, by characteristic p methods, but only for p\geq 5.
Let F be a non-Archimedean local field of residual characteristic p, and {\ell} be a prime number different from p. We consider the local Jacquet-Langlands correspondence between {\ell}-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo {\ell}. More precisely, we show that two integral {\ell}-adic discrete series of GL(m,D) are congruent modulo {\ell} if and only if the same holds for their Jacquet-Langlands transfers to GL(m,D).
Let $K$ be a local non-Archimedean field of positive characteristic and let $L$ be the degree-$n$ unramified extension of $K$. Via the local Langlands and Jacquet-Langlands correspondences, to each sufficiently generic multiplicative character of $L$, one can associate an irreducible representation of the multiplicative group of the central division algebra $D$ of invariant $1/n$ over $K$. In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we prove th…
In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of Levi subgroups of $GSpin$, by constructing Langlands parameters for general generic irreducible admissible representations of $GSpin $ from the one for generic irreducible supercuspidal representations of its Levi subgroups.
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.
