Local Jacquet Langlands Correspondence Research Articles

Affinoids in the Lubin\u2013Tate perfectoid space and simple supercuspidal representations II: wild case

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.

Good reduction of affinoids in the Lubin–Tate curve in even equal characteristic, I

We define affinoids in the Lubin–Tate curve in even equal characteristic, and compute the reductions of them. Each reduction is isomorphic to a smooth affine curve with Artin–Schreier type equation. We expect that the cohomology of the reductions realizes the local Langlands correspondence and local Jacquet–Langlands correspondence for representations of conductor five.

Affinoids in the Lubin–Tate perfectoid space and special cases of the local Langlands correspondence

Following Weinstein, Boyarchenko–Weinstein and Imai–Tsushima, we construct a family of affinoids in the Lubin–Tate perfectoid space and their 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.

Towards an explicit local Jacquet–Langlands correspondence beyond the cuspidal case

We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let $\text{F}$ be a non-Archimedean locally compact field of residue characteristic $p$, and let $\text{G}$ be an inner form of the general linear group $\text{GL}_{n}(\text{F})$ for $n\geqslant 1$. We consider the problem of describing explicitly the local Jacquet–Langlands correspondence $\unicode[STIX]{x1D70B}\mapsto _{\text{JL}}\unicode[STIX]{x1D70B}$ between the complex discrete series representations of $\text{G}$ and $\text{GL}_{n}(\text{F})$, in terms of type theory. We show that the congruence properties of the local Jacquet–Langlands correspondence exhibited by A. Mínguez and the first author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endo-class by the Jacquet–Langlands correspondence can be reduced to the case where the representations $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ are both cuspidal with torsion number $1$. We also give an explicit description of the Jacquet–Langlands correspondence for all essentially tame discrete series representations of $\text{G}$, up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where $\unicode[STIX]{x1D70B}$ and $_{\text{JL}}\unicode[STIX]{x1D70B}$ are both cuspidal.

Good reduction of affinoids in the Lubin–Tate curve in even equal characteristic, II

In the even equal characteristic case, we define a family of affinoids in the Lubin–Tate curve and compute their reductions. The reduction of the affinoid is isomorphic to a smooth affine curve, whose smooth compactification either has high genus or is isomorphic to a supersingular elliptic curve in characteristic two. The first cohomology of the reductions is expected to realize the local Langlands correspondence and the local Jacquet–Langlands correspondence for cuspidal representations of a general linear group of degree two of conductor exponent seven.

PARITY OF THE LANGLANDS PARAMETERS OF CONJUGATE SELF-DUAL REPRESENTATIONS OF AND THE LOCAL JACQUET–LANGLANDS CORRESPONDENCE

We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of$\text{GL}(n)$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.

Affinoids in the Lubin–Tate Perfectoid Space and Simple Supercuspidal Representations I: Tame Case

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.

Local Jacquet–Langlands correspondences for simple supercuspidal representations

We give a description of the local Jacquet-Langlands correspondence for simple supercuspidal representations via type theory. As a consequence, we show that the endo-classes for such representations are invariant under the local Jacquet-Langlands correspondence.

Correspondance de Jacquet–Langlands locale et congruences modulo $$\ell $$ ℓ

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).

Global Jacquet-Langlands Correspondence for Division Algebras in Characteristic $p$

We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero characteristic, we prove that there exists an injective map from the set of automorphic square integrable representations of the multiplicative group of $D$ to the set of automorphic square integrable representations of GL_n(F), compatible at all places with the local Jacquet-Langlands correspondence for unitary representations. We characterize the image of the map. As a consequence we get multiplicity one and strong multiplicity one theorems for the multiplicative group of D.

On non-abelian Lubin–Tate theory for

We analyse the$\text{mod}~p$étale cohomology of the Lubin–Tate tower both with compact support and without support. We prove that there are no supersingular representations in the$H_{c}^{1}$of the Lubin–Tate tower. On the other hand, we show that in$H^{1}$of the Lubin–Tate tower appears the$\text{mod}~p$local Langlands correspondence and the$\text{mod}~p$local Jacquet–Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard–Diamond–Jarvis conjecture which appears naturally in this context.

On the reduction modulo p of representations of a quaternion division algebra over a p-adic field

Let F be a non-Archimedean local field of residue characteristic p. In this paper, we compute the reduction modulo p of irreducible smooth representations of a quaternion division algebra over F and of two-dimensional irreducible smooth representations of the Weil group of F. It turns out that a natural correspondence of mod p representations of the two groups and the composite of the local Langlands correspondence and the local Jacquet–Langlands correspondence are not compatible with the reduction, except in the cases considered by Vignéras.

Geometric approach to the local Jacquet-Langlands correspondence

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for ${\rm GL}(n)$ over a $p$-adic field, under the assumption that the invariant of the division algebra is $1/n$. We use the $\ell$-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 technique nor detailed classification of supercuspidal representations of ${\rm GL}(n)$.

Transfer of Plancherel Measures for Unitary Supercuspidal Representations betweenp-adic Inner Forms

AbstractLetFbe ap-adic field of characteristic 0, and letMbe anF-Levi subgroup of a connected reductiveF-split group such thatGLnifor positive integersrand ni. We prove that the Plancherel measure for any unitary supercuspidal representation ofM(F) is identically transferred underthe local Jacquet–Langlands type correspondencebetweenMand itsF-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups SO4nand Sp4nunderthe local Jacquet–Langlands correspondence. It can be applied to a simply connected simpleF-group of typeE6orE7, and a connected reductiveF-group of typeAn,Bn,CnorDn.

Smooth representations of ${GL}_m(D)$. V: Endo-classes

Let \mathrm F be a locally compact nonarchimedean local field. In this article, we extend to any inner form of \operatorname{GL}_n over \mathrm F , with n\geqslant 1 , the notion of endo-class introduced by Bush­nell and Henniart for \operatorname{GL}_n(\mathrm F) . We investigate the intertwining relations of simple characters of these groups, in particular their preservation properties under transfer. This allows us to associate to any discrete series representation of an inner form of \operatorname{GL}_n(\mathrm F) an endo-class over \mathrm F . We conjecture that this endo-class is invariant under the local Jacquet–Langlands correspondence.

Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations

In this paper we generalize the local Jacquet-Langlands correspondence to 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 Mœglin–Waldspurger and Jacquet–Shalika for GL(n).