Jacquet Langlands Correspondence Research Articles

Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence

Abstract We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p {\mathbb{Q}_{p}} appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases.

On the quantitative variation of congruence ideals and integral periods of modular forms

We prove the conjecture of Pollack and Weston on the quantitative analysis of the level lowering congruence à la Ribet for modular forms of higher weight. It was formulated and studied in the context of the integral Jacquet–Langlands correspondence and anticyclotomic Iwasawa theory for modular forms of weight two and square-free level for the first time. We use a completely different method based on the $$R={\mathbb {T}}$$ theorem established by Diamond–Flach–Guo and Dimitrov and an explicit comparison of adjoint L-values. We briefly discuss arithmetic applications of our main result at the end.

Hodge classes and the Jacquet–Langlands correspondence

Abstract We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\ell $ -adic cohomology is Galois invariant for all $\ell $ .

The spectral p-adic Jacquet–Langlands correspondence and a question of Serre

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

The inertial Jacquet–Langlands correspondence

Abstract We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by two 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.

On an inequality of Bushnell-Henniart for Rankin-Selberg conductors

We prove a division algebra analogue of an ultrametric inequality due to Bushneil—Henniart for Rankin—Selberg conductors. Under the Jacquet—Langlands correspondence, the two versions are equivalent.

ON SOME CONSEQUENCES OF A THEOREM OF J. LUDWIG

AbstractWe prove some qualitative results about thep-adic Jacquet–Langlands correspondence defined by Scholze, in the$\operatorname {\mathrm {GL}}_2(\mathbb{Q}_p )$residually reducible case, using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration, the globalp-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations atpin a nontrivial way, unlike its classical counterpart.

Gamma factors of intertwining periods and distinction for inner forms of GL(n)

Let F be a p-adic field, E be a quadratic extension of F, D be an F-central division algebra of odd index and let θ be the Galois involution attached to E/F. Set H=GL(m,D), G=GL(m,D⊗FE), and let P=MU be a standard parabolic subgroup of G. Let w be a Weyl involution stabilizing M and Mθw be the subgroup of M fixed by the involution θw:m↦θ(wmw). We denote by X(M)w,− the complex torus of w-anti-invariant unramified characters of M. Following the global methods of [30], we associate to a finite length representation σ of M and to a linear form L∈HomMθw(σ,C) a family of H-invariant linear forms called intertwining periods on IndPG(χσ) for χ∈X(M)w,−, which is meromorphic in the variable χ. Then we give sufficient conditions for some of these intertwining periods, namely the open intertwining periods studied in [13], to have singularities. By a local/global method, we also compute in terms of Asai gamma factors the proportionality constants involved in their functional equations with respect to certain intertwining operators. As a consequence, we classify distinguished unitary and ladder representations of G, extending respectively the results of [42] and [26] for D=F, which both relied at some crucial step on the theory of Bernstein-Zelevinsky derivatives. We make use of one of the main results of [12] which in the case of the group G asserts that the Jacquet-Langlands correspondence preserves distinction. Such a result is for essentially square-integrable representations, but our method in fact allows us to use it only for cuspidal representations of G.

A geometric Jacquet\u2013Langlands correspondence for paramodular Siegel threefolds

We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. We give some applications to the Langlands programme: using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet–Langlands correspondence between {text {GSp}}_4 and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of {text {GSp}}_4.

Affine Deligne–Lusztig varieties at infinite level

We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite Deligne–Lusztig construction is isomorphic to an affine Deligne–Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet–Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig’s 1979 conjecture in this setting for minimal admissible characters.

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.

The basis problem revisited

Eichler investigated when there is a basis of a space of modular forms consisting of theta series attached to quaternion algebras, and treated squarefree level. Hijikata, Pizer, and Shemanske completed the solution to Eichler’s basis problem for elliptic modular forms of arbitrary level by tour-de-force trace calculations. We revisit the basis problem using the representation-theoretic perspective of the Jacquet–Langlands correspondence. Our results include: (i) a simpler proof of the solution to the basis problem for elliptic modular forms, which also allows for more flexibility in the choice of quaternion algebra; (ii) a solution to the basis problem for Hilbert modular forms; (iii) a theory of (local and global) new and old forms for quaternion algebras; and (iv) an explicit version of the Jacquet–Langlands correspondence at the level of modular forms, which is a refinement of the Hijikata–Pizer–Shemanske solution to the basis problem. Both (i) and (ii) have practical applications to computing elliptic and Hilbert modular forms. Moreover, (iii) and (iv) are desired for arithmetic applications—to illustrate, we give a simple application to Eisenstein congruences in level p 3 p^3 .

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.

The cohomology of semi-infinite Deligne–Lusztig varieties

Abstract We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h {X_{h}} . Boyarchenko’s two conjectures are on the maximality of X h {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n {1/n} in the case h = 2 {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h {X_{h}} attains its Weil–Deligne bound, so that the cohomology of X h {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ⁢ ( X h ) {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ ↦ H c i ⁢ ( X h ) ⁢ [ θ ] {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

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.

The Jacquet–Langlands correspondence for overconvergent Hilbert modular forms

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 isomorphism between eigenvarieties attached to Hilbert modular forms and those attached to modular forms on a totally definite quaternion algebra over a totally real field of even degree.

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.

A torsion Jacquet-Langlands correspondence

We prove a numerical form of a Jacquet–Langlands correspondence for torsion classes on arithmetic hyperbolic 3-manifolds.