The Weil-Moore anima refines the Weil group into a space with higher homotopy groups to improve its cohomological behavior for number fields.
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Geometrization of the local
20 Pith papers cite this work. Polarity classification is still indexing.
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abstract
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 $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ 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 $\ell$-adic sheaves on $\mathrm{Bun}_G$.
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Proves the Pappas-Rapoport conjecture on canonical integral models of Hodge-type Shimura varieties with quasi-parahoric level at p, shows uniformization by integral local Shimura varieties, and proves the Kisin-Pappas conjecture on local model diagrams.
The authors construct a canonical formal group action on μ-ordinary Igusa varieties for Hodge type Shimura data that integrates Maass-Shimura operators and, via p-adic Fourier theory, yields a unified algebra action encoding both p-adic differential operators and Hecke operators.
For all quasi-split classical p-adic groups and G2, every generic Bernstein block admits a fully faithful categorical functor into ind-coherent sheaves on L-parameter stacks, compatible with parabolic induction and Whittaker data.
Orbital integrals on unitary groups over local fields in positive characteristic converge absolutely.
The p-adic monodromy theorem holds for families of G_K-equivariant vector bundles over the Fargues-Fontaine curve parametrized by algebraic-affinoid Q_p-algebras, enabling classification of line bundles without freeness assumptions.
Constructs the ρ-Fourier transform on L²(G(F)) and the ρ-Schwartz space S_ρ(G(F)) for reductive groups over local fields, proving a large portion of the Braverman-Kazhdan-Ngô conjecture via spectral methods.
Quotient fields of the perfectoid Tate algebra T_n,K^perfd are semi-immediate extensions of K_r1..rl^perfd with l bounded by min(n-ht(m^flat cap coperf),n-1) and at least one ri irrational if the flat intersection is nonzero.
Constructs functorial Igusa stacks for Hodge-type Shimura varieties, yielding a sheaf on Bun_G that controls cohomology and proves compatibility with the semisimple local Langlands correspondence of Fargues-Scholze while establishing torsion vanishing for proper cases.
Anabelomorphic p-adic fields induce isomorphic Langlands parameter stacks, yielding a conjecture relating Fargues-Scholze to anabelomorphy that holds for split tori.
In a perfectoid ring R, I-torsion is I_perfd-almost zero for any ideal I, giving an excision-type decomposition of R along its I-torsion part.
Establishes classicality for Hecke characters in completed cohomology of Hilbert modular varieties under absolute irreducibility and de Rham conditions on Galois representations, giving new cases of the LCFM conjecture for GL2.
Proves relative p-adic monodromy theorem over dense open set and equivalence to Newton polygon constancy near rank-1 points for de Rham local systems, plus extension of conjecture to Newton partition interiors.
Establishes a representability criterion for v-sheaf modifications of formal schemes and applies it to parahoric level structures on local shtukas, yielding local representability of integral models of local Shimura varieties under hyperspecial levels.
J(G) for inner forms G of p-adic GL_n is defined over ar Q_l, admits explicit formulas via reductive centralizers, shares Hochschild homology with C_c^infty(G), and supports rudimentary Hecke compatibility for sheaves on Bun_n.
Any Farey triangle corresponds to a variant of the Colmez-Fontaine fundamental lemma, with the original lemma matching the triangle (1/0, 1/1, 0/1).
New transparent proof of Drinfeld's representability theorem for moduli of p-divisible groups with extra actions, plus detailed presentation of the moduli space and formal model of the p-adic symmetric space.
A result is established about the non-generic cohomology of certain compact unitary Shimura varieties for good p, extending Boyer's work via a different approach in the Fargues-Scholze context.
Proves a deformation theorem for prismatic higher (G,μ)-displays over quasi-syntomic rings, extends p-divisible group classification, and relates the display stack to integral local Shimura varieties.
A survey paper presents the Geometric Langlands correspondence informally as an algebraic spectral theorem for automorphic sheaves and a blueprint for studying nonabelian symmetry.
citing papers explorer
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Weil-Moore anima
The Weil-Moore anima refines the Weil group into a space with higher homotopy groups to improve its cohomological behavior for number fields.
-
On a conjecture of Pappas and Rapoport
Proves the Pappas-Rapoport conjecture on canonical integral models of Hodge-type Shimura varieties with quasi-parahoric level at p, shows uniformization by integral local Shimura varieties, and proves the Kisin-Pappas conjecture on local model diagrams.
-
$p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties
The authors construct a canonical formal group action on μ-ordinary Igusa varieties for Hodge type Shimura data that integrates Maass-Shimura operators and, via p-adic Fourier theory, yields a unified algebra action encoding both p-adic differential operators and Hecke operators.
-
A generic categorical local Langlands correspondence for quasi-split reductive groups
For all quasi-split classical p-adic groups and G2, every generic Bernstein block admits a fully faithful categorical functor into ind-coherent sheaves on L-parameter stacks, compatible with parabolic induction and Whittaker data.
-
Convergence of orbital integrals on unitary groups in positive characteristic
Orbital integrals on unitary groups over local fields in positive characteristic converge absolutely.
-
The $p$-adic monodromy theorem over algebraic-affinoid algebras
The p-adic monodromy theorem holds for families of G_K-equivariant vector bundles over the Fargues-Fontaine curve parametrized by algebraic-affinoid Q_p-algebras, enabling classification of line bundles without freeness assumptions.
-
The $\rho$-Fourier transform
Constructs the ρ-Fourier transform on L²(G(F)) and the ρ-Schwartz space S_ρ(G(F)) for reductive groups over local fields, proving a large portion of the Braverman-Kazhdan-Ngô conjecture via spectral methods.
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Abhyankar valuations, Pr\"ufer-Manis valuations, and perfectoid Tate algebras
Quotient fields of the perfectoid Tate algebra T_n,K^perfd are semi-immediate extensions of K_r1..rl^perfd with l bounded by min(n-ht(m^flat cap coperf),n-1) and at least one ri irrational if the flat intersection is nonzero.
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Igusa Stacks and the Cohomology of Shimura Varieties
Constructs functorial Igusa stacks for Hodge-type Shimura varieties, yielding a sheaf on Bun_G that controls cohomology and proves compatibility with the semisimple local Langlands correspondence of Fargues-Scholze while establishing torsion vanishing for proper cases.
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The Categorical Local Langlands Correspondence and Anabelomorphy
Anabelomorphic p-adic fields induce isomorphic Langlands parameter stacks, yielding a conjecture relating Fargues-Scholze to anabelomorphy that holds for split tori.
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Decomposition Theorem for Perfectoid Rings along General Ideals
In a perfectoid ring R, I-torsion is I_perfd-almost zero for any ideal I, giving an excision-type decomposition of R along its I-torsion part.
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Classicality for Hilbert modular forms
Establishes classicality for Hecke characters in completed cohomology of Hilbert modular varieties under absolute irreducibility and de Rham conditions on Galois representations, giving new cases of the LCFM conjecture for GL2.
-
p-adic Hodge theory of de Rham local systems, I: Newton polygon and monodromy
Proves relative p-adic monodromy theorem over dense open set and equivalence to Newton polygon constancy near rank-1 points for de Rham local systems, plus extension of conjecture to Newton partition interiors.
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Relative representability and parahoric level structures
Establishes a representability criterion for v-sheaf modifications of formal schemes and applies it to parahoric level structures on local shtukas, yielding local representability of integral models of local Shimura varieties under hyperspecial levels.
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On Braverman-Kazhdan's asymptotic Hecke algebra for inner forms of $\mathrm{GL}_n$
J(G) for inner forms G of p-adic GL_n is defined over ar Q_l, admits explicit formulas via reductive centralizers, shares Hochschild homology with C_c^infty(G), and supports rudimentary Hecke compatibility for sheaves on Bun_n.
-
Modular variants of p-adic fundamental sequence
Any Farey triangle corresponds to a variant of the Colmez-Fontaine fundamental lemma, with the original lemma matching the triangle (1/0, 1/1, 0/1).
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On Drinfeld's representability theorem
New transparent proof of Drinfeld's representability theorem for moduli of p-divisible groups with extra actions, plus detailed presentation of the moduli space and formal model of the p-adic symmetric space.
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On the non-generic part of cohomology of compact unitary Shimura varieties of signature $(1,n)$
A result is established about the non-generic cohomology of certain compact unitary Shimura varieties for good p, extending Boyer's work via a different approach in the Fargues-Scholze context.
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Deformations of Prismatic Higher $(G,\mu)$-Displays over Quasi-Syntomic Rings
Proves a deformation theorem for prismatic higher (G,μ)-displays over quasi-syntomic rings, extends p-divisible group classification, and relates the display stack to integral local Shimura varieties.
-
What is the Geometric Langlands Correspondence about?
A survey paper presents the Geometric Langlands correspondence informally as an algebraic spectral theorem for automorphic sheaves and a blueprint for studying nonabelian symmetry.