Moduli of Langlands Parameters

Dat, Jean-François; Helm, David; Kurinczuk, Robert; Moss, Gilbert

doi:10.48550/arxiv.2009.06708

Cited by 10 publications

(31 citation statements)

References 26 publications

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“…Let us discuss the other side of the Langlands correspondence, namely (the stack of) Lparameters. This has been previously done by Dat-Helm-Kurinczuk-Moss [DHKM20] and Zhu [Zhu20]. One wants to define a scheme whose Λ-valued points, for a Z -algebra Λ, are the continuous 1-cocycles ϕ : W E → G(Λ).…”

Section: I8 the Stack Of L-parametersmentioning

confidence: 99%

“…The point here is that the inertia subgroup of W E has a Z -factor, and this can map in interesting ways to Λ when making this definition. To prove the theorem, following [DHKM20] and [Zhu20] we define discrete dense subgroups W ⊂ W E /P by discretizing the tame inertia, and the restriction Z 1 (W E /P, G) → Z 1 (W, G) is an isomorphism, where the latter is clearly an affine scheme.…”

Section: I8 the Stack Of L-parametersmentioning

confidence: 99%

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Geometrization of the local Langlands correspondence

Fargues,

Scholze

2021

Preprint

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Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues-Fontaine curve. In particular, we define a category of -adic sheaves on the stack BunG of G-bundles on the Fargues-Fontaine curve, prove a geometric Satake equivalence over the Fargues-Fontaine curve, and study the stack of L-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define L-parameters associated with irreducible smooth representations of G(E), a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of L-parameters on the category of -adic sheaves on BunG. ContentsChapter I. Introduction I.1. The local Langlands correspondence I.2. The big picture I.3. The Fargues-Fontaine curve I.4. The geometry of Bun G I.5. -adic sheaves on Bun G I.6. The geometric Satake equivalence I.7. Cohomology of moduli spaces of shtuka I.8. The stack of L-parameters I.9. Construction of L-parameters I.10. The spectral action I.11. The origin of the ideas I.12. Acknowledgments I.13. Notation Chapter II. The Fargues-Fontaine curve and vector bundles V.1. Classifying stacks V.2. Étale sheaves on strata V.3. Local charts V.4. Compact generation V.5. Bernstein-Zelevinsky duality V.6. Verdier duality V.7. ULA sheaves Chapter VI. Geometric Satake VI.1. The Beilinson-Drinfeld Grassmannian VI.2. Schubert varieties VI.3. Semi-infinite orbits VI.4. Equivariant sheaves VI.5. Affine flag variety VI.6. ULA sheaves VI.7. Perverse Sheaves VI.8. Convolution VI.9. Fusion VI.10. Tannakian reconstruction VI.11. Identification of the dual group VI.12. Cartan involution Chapter VII. D (X) VII.1. Solid sheaves VII.2. Four functors VII.3. Relative homology VII.4. Relation to D ét VII.5. Dualizability VII.6. Lisse sheaves VII.7. D lis (Bun G ) Chapter VIII. L-parameter VIII.1. The stack of L-parameters VIII.2. The singularities of the moduli space VIII.3. The coarse moduli space VIII.4. Excursion operators VIII.5. Modular representation theory Chapter IX. The Hecke action IX.1. Condensed ∞-categories IX.2. Hecke operators IX.3. Cohomology of local Shimura varieties IX.4. L-parameter IX.5. The Bernstein center IX.6. Properties of the correspondence IX.7. Applications to representations of G(E) CONTENTS Chapter X. The spectral action X.1. Rational coefficients X.2. Elliptic parameters X.3. Integral coefficients Bibliography CHAPTER I

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Section: I8 the Stack Of L-parametersmentioning

confidence: 99%

Section: I8 the Stack Of L-parametersmentioning

confidence: 99%

Geometrization of the local Langlands correspondence

Fargues,

Scholze

2021

Preprint

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“…], can be reformulated in a different, more geometric, manner. Let X Ĝ be the algebraic stack over L of n-dimensional continuous L-representations of W E , defined and studied in [8, VIII.1] or [3], [20]. The algebraic stack X Ĝ comes with a map f : X Ĝ → [Spec(L)/ Ĝ].…”

Section: 4mentioning

confidence: 99%

Averaging functors in Fargues' program for GL_n

Anschütz,

Bras

2021

Preprint

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We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues' program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for GLn. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of GL 2 are irreducible. We also attach to any irreducible ℓ-adic Weil representation of degree n an Hecke eigensheaf on Bunn, and show, using the local Langlands correspondence and recent results of Hansen and Kaletha-Weinstein, that it satisfies most of the requirements of Fargues' conjecture for GLn. Contents 1. Introduction 1.1. Acknowledgements 2. Notation 3. Geometrization of local Langlands: recollection 3.1. The stack of vector bundles on the Fargues-Fontaine curve 3.2. The category of lisse ℓadic sheaves 3.3. Statement of Fargues' conjecture for GL n 3.4. Fargues-Scholze's construction of the spectral action 4. Constant term and Eisenstein functors 4.1. Constant terms and twisting 4.2. Geometrically cuspidal representations 5. Averaging functors 5.1. The definition of the averaging functors 5.2. Averaging functors and constant terms 5.3. Averaging functors and twisting by characters 5.4. The averaging functors of rank 1ocal systems 5.5. Averaging functors via the spectral action 5.6. Averaging functors on D lis 6. Application to irreducibility of the Fargues-Scholze parameters if n = 2 7. Application to Fargues' conjecture in the irreducible case 7.1. Generalities on Hecke eigensheaves for irreducible representations 7.2. The spectral action and the definition of Aut L 7.3. Generalities on the spectral action of k(L) i 7.4. Computation of one stalk of Aut L and conclusion References Date: April 13, 2021.1 Depending on G one has to put some assumption on ℓ. However, for G = GLn each ℓ works.

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“…If we let A/Z ℓ be any Z ℓ -algebra endowed with a topology given by writing A = colim A ′ ⊂A A ′ , where A ′ is a finitely generated Z ℓ -module with its ℓ-adic topology, then we can defined a moduli space, denoted Z 1 (W Qp , Ĝ), over Z ℓ , whose A-points are the continuous 1-cocyles W Qp → Ĝ(A) with respect to the natural action of W Qp on Ĝ(A). This defines a scheme considered in [Dat+20] and [Zhu20] which, by [FS21, Theorem I.9.1], can be written as a union of open and closed affine subschemes Z 1 (W Qp /P, Ĝ) as P runs through subgroups of wild inertia of W E , where each Z 1 (W Qp /P, Ĝ) is a flat local complete intersection over Z ℓ of dimension dim(G). This allows us to consider the Artin stack quotient [Z 1 (W Qp , Ĝ)/ Ĝ], where Ĝ acts via conjugation.…”

Section: Qpmentioning

confidence: 99%

Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands

Hamann¹

2021

Preprint

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Given a prime p, a finite extension L/Q p , a connected p-adic reductive group G/L, and a smooth irreducible representation π of G(L), Fargues-Scholze [FS21] recently attached a semisimple Weil parameter to such π, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For G = GL n and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein [HKW21] show that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart [Hen00; HT01]. We verify a similar compatibility for G = GSp 4 and its unique non-split inner form G = GU 2 (D), where D is the quaternion division algebra over L, assuming that L/Q p is unramified and p > 2. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono [GT11; GT14]. Analogous to the case of GL n and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to GSp 4 , using basic uniformization of abelian type Shimura varieties due to Shen [She17], combined with various global results of Kret-Shin [KS16] and Sorensen [Sor10] on Galois representations in the cohomology of global Shimura varieties associated to inner forms of GSp 4 over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen [Han20], to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process. LINUS HAMANN Applications 48 References 51 AcknowledgementsIt is a pleasure to thank my advisor David Hansen for giving me this project and for numerous suggestions and ideas related to it, as well as Eric Chen, Arthur-César Le-Bras, Yoichi Mieda, Jack Sempliner, and Zhiyu Zhang for some nice conversations related to this work. Special thanks also go to Peter Scholze for sharing with me the draft of [FS21], pointing out some errors, and generously initiating me in these ideas during my masters thesis, Sug-Woo Shin for pointing out some errors in section 5, and the MPIM Bonn for their hospitality during part of the completion of this project.

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Moduli of Langlands Parameters

Cited by 10 publications

References 26 publications

Geometrization of the local Langlands correspondence

Geometrization of the local Langlands correspondence

Averaging functors in Fargues' program for GL_n

Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands

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