We prove the formal degree conjecture for simple supercuspidal representations of symplectic groups and quasi-split even special orthogonal groups over a p-adic field, under the assumption that p is odd. The essential part is to compute the Swan conductor of the exterior square of an irreducible local Galois representation with Swan conductor 1. It is carried out by passing to the equal characteristic local field and using the theory of Kloosterman sheaves.
We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein's work proving the formal degree conjecture for regular supercuspidal representations. The main difference between our work and Schwein's work is that in non-singular case, the Deligne--Lusztig representations can be reducible, and the $S$-groups are not necessary abelian. Therefore, we have to compare the dimensions of irreducible constituents of the Deligne--Lusztig representations and the dimensions of irreducible representations of $S$-groups.
Supercuspidal representations are usually infinite-dimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its $L$-parameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second, using the first, is to verify that Kaletha's regular supercuspidal $L$-packets satisfy the conjecture.
The formal degree conjecture and the root number conjecture are verified with respect to supercuspidal representations of $Sp_{2n}(F)$ and $L$-parameters associated with tamely ramified extension $K/F$ of degree $2n$. The supercuspidal representation is constructed as a compact induction from an irreducible unitary representation of the hyper special compact group $Sp_{2n}(O_F)$, which is explicitly constructed, based upon the general theory developed by the author, by $K$ and certain character $θ$ of the multiplicative group $K^{\times}$. $L$-parameter is constructed by the data $\{K,θ\}$ by …
Based upon the general theory, developed by the author, on the parametrization of the irreducible representations of the hyper special compact groups corresponding to the regular adjoint orbit, supercuspidal representations of $SL_n(F)$ are explicitly constructed for which the formal degree conjecture and the root number conjecture are verified with respect to certain $L$-parameter defined, by means of Kaletha, that is, the local Langlands correspondence of tori and the Langlands-Schelstad procedure, by the data parametrizing the irreducible representations of the hyper special compact subgrou…
In this article, we study Prasad's conjecture for regular supercuspidal representations based on the machinery developed by Hakim and Murnaghan to study distinguished representations, and the fundamental work of Kaletha on parameterization of regular supercuspidal representations. For regular supercuspidal representations, we give some new interpretations of the numerical quantities appearing in Prasad's formula, and reduce the proof to the case of tori. The proof of Prasad's conjecture then reduces to a comparison of various quadratic characters appearing naturally in the above process. We al…
Let $F$ be a non-archimedean local field of residue characteristic $p\neq 2$. Let $G$ be a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Yu constructed types which are called tame supercuspidal types and conjectured that Hecke algebras associated with these types are isomorphic to Hecke algebras associated with depth-zero types of some twisted Levi subgroups of $G$. In this paper, we prove this conjecture. We also prove that the Hecke algebra associated with a regular supercuspidal type is isomorphic to the group algebra of a certain abelian group.
Let G be the unramified unitary group in three variables defined over a p-adic field F of odd resudual characteristic. Gelbart, Piatetski-Shapiro and Baruch attached zeta integrals of Rankin-Selberg type to irreducible generic representations of G. In this paper, we formulate a conjecture on L- and epsilon-factors defined through zeta integrals in terms of local newforms for G, which is an analogue of the result by Casselman and Deligne for GL(2). We prove our conjecture for the generic supercuspidal representations of G.
We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of the reductions of the formal models realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for simple supercuspidal representations in the case where the dimension of Galois representations is prime to the residue characteristic. The reductions of the formal models are isomorphic to the perfections of Artin-Schreier varieties associated to quadratic forms.
The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of $G_2$ over a $p$-adic field, one can associate a generic supercuspidal irreducible representation of either $PGSp_6$ or$PGL_3$. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of $G_2$ and other representations of $PGSp_6$ and $PGL_3$. This correspondence arises from theta correspondences in $E_6$ and $E_7$, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of g…
We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of their reductions realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for the simple supercuspidal representations. The reductions of the formal models are isomorphic to the perfections of some Artin-Schreier varieties, whose cohomology realizes primitive Galois representations. We show also the Tate conjecture for Artin-Schreier varieties associated to quadratic forms.
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.
We will give an explicit construction of irreducible suparcuspidal representations of the special linear group over a non-archimedean local field and will speculate its Langlands parameter by means of verifying the Hiraga-Ichino-Ikeda formula of the formal degree of the supercuspidal representations.
In this work, we explicitly compute a certain family of twisted gamma factors of a simple supercuspidal representation $π$ of a $p$-adic odd orthogonal group. These computations, together with analogous computations for general linear groups carried out in previous work with Liu, allow us to give a prediction for the Langlands parameter of $π$. If we assume the "depth-preserving conjecture", we prove that our prediction is correct if $p$ is sufficiently large.
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.
Kottwitz's conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze's more general spaces of local shtukas. Using a new Lefschetz-Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that for an irreducible smooth representation of an inner form of $\math…
As a consequence of his numerical local Langlands correspondence for $GL(n)$, Henniart deduced the following theorem: If $F$ is a nonarchimedean local field and if $π$ is an irreducible admissible representation of $GL(n,F)$, then, after a finite sequence of cyclic base changes, the image of $π$ contains a vector fixed under an Iwahori subgroup. This result was indispensable in all proofs of the local Langlands correspondence. Scholze later gave a different proof, based on the analysis of nearby cycles in the cohomology of the Lubin-Tate tower. Let $G$ be a reductive group over $F$. Assuming a…
We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm{GL_n}$ and to show local-global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms, $\mathrm{Mant}_{b, μ}$, of Grothendieck groups of representations constructed from the cohomology of the above spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm{Mant}_{b, μ}(ρ)$ for $ρ$ a s…
The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.
In this paper, we prove a kernel formula of Bessel functions attached to irreducible smooth supercuspidal representations of p-adic $GL(n)$. We also show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell. As an application we give a proof of the local converse conjecture of Jacquet.
Let $G$ be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of $G$, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of $\operatorname{Bun}_G$. It is independent of the work of Lafforgue on the global Langlands correspondence.
In 1980 Zelevinsky introduced commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss--Leclerc--Schröer condition that occurs in the conjectural characterization of $\square$-irreducible representations. We verify some special cases of the new conjectu…
We give a general proof of Shahidi's tempered L-function conjecture, which has previously been known in all but one case. One of the consequences is the standard modules conjecture for p-adic groups, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible and the inducing data generic. We have also included the result that every generic tempered representation of a p-adic group is a sub-representation of a representation parabolically induced from a generic supercuspidal representation with a non-negative real central character.
Let $\ell$ be a prime and let $q$ be a prime power not divisible by $\ell$. Put $G=\mathrm{GL}_n(\mathrm{F}_q)$ and fix an irreducible cuspidal representation, $\barπ$, of $G$ over a sufficiently large finite field, $k$, of characteristic $\ell$ such that $\barπ$ is not supercuspidal. We compute the $\mathrm{W}(k)[G]$-endomorphism ring of the projective envelope of $\barπ$ under the assumption that $\ell>n$. Our computations provide evidence for a conjecture of Helm relating the Bernstein center to the deformation theory of Galois representations.
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.
We explicitly compute the adjoint L-function of those L-packets of representations of the group GSp(4) over a p-adic field of characteristic zero that contain non-supercuspidal representations. As an application we verify a conjecture of Gross-Prasad and Rallis in this case. The conjecture states that the adjoint L-function has a pole at s=1 if and only if the L-packet contains a generic representation.
Let F be a non-archimedean local field, of characteristic 0. Let V be a finite dimensional vector space over F and q be a non-degenerate quadratic form on V. Denote d the dimension of V and G=SO(d) the special orthogonal group of (V,q). Let v_{0}\in V such that q(v_{0})\not=0, denote W the subspace of V orthogonal to v_{0} and H=SO(d-1) the special orthogonal group of W. Let π, resp. σ, an admissible irreducible representation of G(F), resp. H(F). Denote m(σ,π) the dimension of the complex space Hom_{H(F)}(π_{| H(F)},σ). By a theorem of Aizenbud, Gourevitch, Rallis and Schiffmann, we know that…
The conjecture stated by Carayol in [{\em Non-abelian Lubin-Tate theory.} Automorphic forms, Shimura varieties and $L$-functions, vol II: 15--39, Academic Press,1990] predicted that the {\em supercuspidal part} of the l-adic cohomology of the moduli spaces classifying certain formal groups introduced by Drinfeld would provide a simultaneaous "realization" of Langlands and Jacquet-Langlands local correspondences for {\em supercuspidal} representations of $GL_d$. By works of Harris, Boyer, Harris-Taylor and Hausberger, this conjecture is now proved. On another hand, recent works by Boyer and Fal…
We compute the universal deformations of cuspidal representations $π$ of $\GL_2(F)$ over an algebraically closed field of characteristic $l$, where $F$ is a local field of residue characteristic $p$ not equal to $l$. When $π$ is supercuspidal there is an irreducible, two-dimensional representation $ρ$ of $G_F$ that corresponds to $π$ by the mod $l$ local Langlands correspondence of Vign{é}ras; we show there is a natural isomorphism between the universal deformation rings of $π$ and $ρ$ that induces the usual local Langlands correspondence on characteristic zero points. Our work establishes cer…
Let $O$ be the ring of power series in one variable over a finite field, with $K$ its fraction field. We introduce the notion of a "formal $K$-vector space"; this is a certain kind of $K$-vector space object in the category of formal schemes. This concept runs parallel to the established notion of a formal $O$-module, but in many ways formal $K$-vector spaces are much simpler objects. Our main result concerns the Lubin-Tate tower, which plays a vital role in the local Langlands correspondence for $GL_n(K)$. Let $A_m$ be the complete local ring parametrizing deformations of a fixed formal $O$-m…
The Local Converse Problem is to determine how the family of the local gamma factors $γ(s,π\timesτ,ψ)$ characterizes the isomorphism class of an irreducible admissible generic representation $π$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $τ$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, w…
We study the geometry and cohomology of the (generic fibres) of formal deformation schemes of one-dimensional formal modules of finite height. By the work of Boyer (in mixed characterististic) and Harris and Taylor, the l-adic etale cohomology of these spaces realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence for supercuspidal representations is realized by…
In this paper we study quantitative aspects of trace characters $Θ_π$ of reductive $p$-adic groups when the representation $π$ varies. Our approach is based on the local constancy of characters and we survey some other related results. We formulate a conjecture on the behavior of $Θ_π$ relative to the formal degree of $π$, which we are able to prove in the case where $π$ is a tame supercuspidal. The proof builds on J.-K.~Yu's construction and the structure of Moy-Prasad subgroups.
Let H be a connected reductive group over an algebraically closed field of characteristic zero, and let G be an abstract group. In this note we show that every homomorphism from the Grothendieck semiring of H to that of G which maps irreducible representations to irreducibles, comes from a group homomorphism from G to H. We also connect this result with the Langlands conjectures.
We generalize the work of M. Harris and R. Taylor on the local Langlands correspondence for the linear group over $\mathbb{Q}_p$. We prove some cases of the Kottwitz conjectures for the supercuspidal part of the compactly supported $\ell$-adic cohomology of Rapoport-Zink rigid-analytic spaces. This means that we decompose this part in terms of local Langlands correspondences. In particular, we prove the first case of a non-abelian reciprocity law constructed geometrically and associated to groups other than inner forms of the linear group. More precisely, we do this for the unramified unitary …
In this paper we characterize irreducible generic representations of $\SO_{2n+1}(k)$ where $k$ is a $p$-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of $\SO_{2n+1}({\Bbb A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem);and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of $\SO_{2n+1}(k)$.
In a recent paper, Gross and Reeder study arithmetic properties of discrete Langlands parameters for semi-simple p-adic groups and conjecture that a special class of these -- the simple wild parameters -- should correspond to L-packets consisting of simple supercuspidal representations. We provide a construction of this correspondence and show that the simple wild L-packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open …
Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne group of F. This bijection is required to satisfy certain natural compatibilities, of which the most important is compatibility with local functional equations (preservation of L and epsilon factors of pairs). It is enough to construct a bijection with these properties between supercuspidal representations of GL(n,F) and n-dimensional irreducible representa…
Let σ\otimesπ be a supercuspidal representation of SO(2n) \times GL(2n) over a p-adic field with π selfdual, where SO(2n) stands for a quasisplit even special orthogonal group. In order to study its normalized parabolic induction to SO(6n), Goldberg and Shahidi defined a pairing R between the matrix coefficients of σ and π which controls the residue of the standard intertwining operator. The elliptic part of R is conjectured to be related to twisted endoscopic transfer. Based on Arthur's endoscopic classification and Spallone's improvement of Goldberg-Shahidi program, we will verify some of th…
In this article, we consider the links between parabolic induction and the local Langlands correspondence. We enunciate a conjecture about the (enhanced) Langlands parameters of supercuspidal representation of split reductives $p$-adics groups. We are able to verify this in those known cases of the local Langlands correspondence for linear groups and classical groups. Furthermore, in the case of classical groups, we can construct the "cuspidal support" of an enhanced Langlands parameter and get a decomposition of the set of enhanced Langlands parameters a la Bernstein. We check that these cons…
We prove $p$-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a $p$-adic field $K$ or its ring of integers $R$, respectively. In particular, we show that the rigidity results for algebraic functions underlying the so-called Manin-Mumford Conjecture generalize to suitable $p$-adic analytic functions. In the formal setting, this approach leads us to uncover purely $p$-adi…
We provide an explicit construction of the local Langlands correspondence for general tamely-ramified reductive p-adic groups and a class of wildly ramified Langlands parameters. Furthermore, we verify that our construction satisfies the expected properties of such a correspondence. More precisely, we show that each L-packet we construct admits a parameterization in terms of the Langlands dual group, contains a unique generic element for a fixed Whittaker datum, satisfies the formal degree conjecture, is compatible with twists by central and cocentral characters, provides a stable virtual char…
Lapid and Mao formulated a conjecture on an explicit formula of Whittaker Fourier coefficients of automorphic forms on quasi-split classical groups and metaplectic groups as an analogue of Ichino-Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this paper, we study even unitary group case. Indeed, we prove this local identity over $p$-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prov…
We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogues of Fourier-Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla's conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.
We give a new local proof of the Breuil-Mézard conjecture in the case of a reducible representation of the absolute Galois group of $\mathbb{Q}_p$, $p>2$, that has scalar semi-simplification, via a formalism of Paškūnas.
