Langlands Correspondence Research Articles

Quantum Hall Effect and Langlands Program

Recent advances in the Langlands program shed light on a vast area of modern mathematics from an unconventional viewpoint, including number theory, gauge theory, representation, knot theory and etc. By applying to physics, these novel perspectives endow with a unified account of the (integer/ fractional) quantum Hall effect. The plateaus of the Hall conductance are described by Hecke eigensheaves of the geometric Langlands correspondence. Especially, the particle-vortex duality, which is explained byS-duality of Chern–Simons theory, corresponds to the Langlands duality in Wilson and Hecke operators. Moreover the Langlands duality in the quantum group associated with the Hamiltonian describes fractal energy spectrum structure, know as Hofstadter’s butterfly. These results suggest that the Langlands program has many physically realistic meanings.

Globalization of supercuspidal representations over function fields and applications

Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p > 0 . Using Poincaré series, we globalize supercuspidal representations of H_F in such a way that we have control over ramification at all other places, and such that the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved. In combination with the work of Vincent Lafforgue on the global Langlands correspondence, we present some applications, such as the stability of Langlands–Shahidi \gamma -factors and the local Langlands correspondence for classical groups.

Integrating quantum groups over surfaces

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the (0,1,2)-dimensional part of Crane–Yetter–Kauffman four-dimensional TFTs associated to modular categories. Starting from modules for the Drinfeld–Jimbo quantum group U q ( g ) we obtain in this way an aspect of topologically twisted four-dimensional N = 4 super Yang–Mills theory, the setting introduced by Kapustin–Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of G-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to U q ( g ) , and from the punctured torus we recover the algebra of quantum differential operators associated to U q ( g ) . From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum D-modules.

Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces

Rapoport-Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their $l$-adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the $l$-adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport-Zink spaces which are Hodge type and Hodge-Newton reducible. The main strategy is to embed such a Rapoport-Zink space into an appropriate space of EL type, for which the conjecture is already known to hold by the work of Mantovan.

An algorithm for computing the reduction of 2-dimensional crystalline representations of Gal(ℚ¯p/ℚp)

We describe an algorithm to compute the reduction modulo [Formula: see text] of a crystalline Galois representation of dimension [Formula: see text] of [Formula: see text] with distinct Hodge–Tate weights via the semi-simple modulo [Formula: see text] Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE.

Shelstad's character identity from the point of view of index theory

Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding L-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of K-theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program.

Hofstadter’s butterfly confirms a new tie between physics and mathematics

A researcher finds novel connections between Hofstadter’s butterfly and the mathematical conjecture known as the Langlands program.

On multiplicity in restriction of tempered representations of p-adic groups

We establish an equality between two multiplicities: one in the restriction of tempered representations of a p-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $$\mathcal {S}$$ -groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered L-packets. This provides a formula of the multiplicity for p-adic groups by means of dimensions of irreducible representations of their $$\mathcal {S}$$ -groups.

Transfer of Representations and Orbital Integrals for Inner Forms of GLn

AbstractWe characterize the Local Langlands Correspondence (LLC) for inner forms of GLn via the Jacquet–Langlands Correspondence (JLC) and compatibility with the Langlands Classification. We show that LLC satisfies a natural compatibility with parabolic induction and characterize LLC for inner forms as a unique family of bijections Π(GLr(D)) → Φ(GLr(D)) for each r, (for a fixed D), satisfying certain properties. We construct a surjective map of Bernstein centers ℨ(GLn(F)) → ℨ(GLr(D)) and show this produces pairs of matching distributions in the sense of Haines. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of GLr(D), and thereby produce many explicit pairs of matching functions.

WHAT IS ...the Langlands Program?

WHAT IS ...the Langlands Program?

Hofstadter’s butterfly and Langlands duality

We address the Hofstadter problem on a two-dimensional square lattice system. We propose a novel perspective on its mathematical structure of the corresponding tight-binding Hamiltonian from a viewpoint of the Langlands duality, a mathematical conjecture relevant to a wide range of the modern mathematics including number theory, solvable systems, representations, and geometry. It is known that the Hamiltonian can be algebraically written by means of the quantum group Uq(sl2). We claim that Hofstadter’s fractal is deeply related with the Langlands duality of the quantum group. In addition, from this perspective, the existence of the corresponding elliptic curve expression interpreted from the tight-binging Hamiltonian implies a more fascinating connection with the Langlands program and quantum geometry.

On the asymptotic Fermat’s last theorem over number fields

Let K be a number field, S be the set of primes of K above 2 and T the subset of primes above 2 having inertial degree 1. Suppose that T \ne \emptyset , and moreover, that for every solution (\lambda,\mu) to the S -unit equation \lambda+\mu=1, \quad \lambda,~\mu \in \mathcal O_S^\times, there is some \mathfrak P \in T such that max \{ \nu_\mathfrak P(\lambda),\nu_\mathfrak P(\mu)\} \le 4 \nu_\mathfrak P(2) . Assuming two deep but standard conjectures from the Langlands programme, we prove the asymptotic Fermat's last theorem over K : there is some B_K such that for all prime exponents p > B_K the only solutions to x^p+y^p+z^p=0 with x , y , z \in K satisfy xyz=0 . We deduce that the asymptotic Fermat's last theorem holds for imaginary quadratic fields \mathbb Q(\sqrt{-d}) with -d \equiv 2, 3 (mod) 4) squarefree.

Sur les -blocs de niveau zéro des groupes -adiques

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

Langlands correspondence for isocrystals and the existence of crystalline companions for curves

In this paper, we show the Langlands correspondence for isocrystals on curves, which asserts the existence of crystalline companions in the case of curves. For the proof we generalize the theory of arithmetic D \mathscr {D} -modules to algebraic stacks whose diagonal morphisms are finite. Finally, combining with methods of Deligne and Drinfeld, we show the existence of an “ ℓ \ell -adic companion” for any isocrystal on a smooth scheme of any dimension under the assumption of a Bertini-type conjecture.

On Fermat\u2019s equation over some quadratic imaginary number fields

Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat’s Last Theorem over {mathbb {Q}}(i). Under the same assumption, we also prove that, for all prime exponents p ge 5, Fermat’s equation a^p+b^p+c^p=0 does not have non-trivial solutions over {mathbb {Q}}(sqrt{-2}) and {mathbb {Q}}(sqrt{-7}).

Reductions of Galois representations of slope 1

We compute the reductions of irreducible crystalline two-dimensional representations of GQp of slope 1, for primes p≥5, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the extension obtained is peu or très ramifiée, in the relevant reducible non-semisimple cases. The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences, and involves a detailed study of the reductions of both the standard and non-standard lattices in certain p-adic Banach spaces.

Picard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for SL2

Let G be a reductive group and let Bun_G denote the moduli stack of G-bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of Bun_G due to V. Drinfeld (unpublished), which we refer to as the Drinfeld-Lafforgue-Vinberg compactification. For G=GL_n certain smooth open substacks of this compactification have already appeared in the work of Drinfeld and of L. Lafforgue on the Langlands correspondence for function fields. The Drinfeld-Lafforgue-Vinberg compactification is however already singular for G=SL_2; questions about its singularities arise naturally in the geometric Langlands program, and form the topic of the present article. Drinfeld's definition of the compactification for a general reductive group G relies on the Vinberg semigroup of G, and will be given in [Sch]. In the present paper we focus on the case G=SL_2. In this case the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of SL_2-bundles. We study the singularities of this degeneration via the weight-monodromy theory of its nearby cycles: We give an explicit description of the nearby cycles sheaf in terms of certain novel perverse sheaves which we call "Picard-Lefschetz oscillators" and which govern the singularities of the degeneration. We then use this description to determine its intersection cohomology sheaf and other invariants of its singularities. We also discuss the relationship of our results for G=SL_2 with the miraculous duality of Drinfeld and Gaitsgory in the geometric Langlands program, as well as two applications of our results to the classical theory: To Drinfeld's and Wang's "strange" invariant bilinear form on the space of automorphic forms; and to the categorification of the Bernstein asymptotics map studied by Bezrukavnikov and Kazhdan as well as by Sakellaridis and Venkatesh.

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

For any reductive groupGGover a global function field, we use the cohomology ofGG-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence forGGin the “automorphic to Galois” direction. Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet–Langlands correspondence

We study arithmetic intersections on 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.

On the unicity of types in special linear groups

Let F be a non-archimedean local field. We show that any representation of a maximal compact subgroup of mathbf {SL}_N(F) which is typical for an essentially tame supercuspidal representation must be induced from a Bushnell–Kutzko maximal simple type. From this, we explicitly count and describe the conjugacy classes of such typical representations, and give an explicit description of an inertial Langlands correspondence for essentially tame irreducible N-dimensional projective representations of the Weil group of F.