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Documents  Fargues, Laurent | enregistrements trouvés : 6

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Research talks;Algebraic and Complex Geometry;Number Theory

(joint work with Peter Scholze) In our joint work with Scholze we need to give a meaning to statements like "the stack of principal G-bundles on the curve is smooth of dimension 0" and construct "smooth perfectoid charts on it". The problem is that in the perfectoid world there is no infinitesimals and thus no Jacobian criterion that would allow us to define what is a smooth morphism. The good notion in this setting is the one of a cohomologically smooth morphism, a morphism that satisfies relative Poincaré duality. I will explain a Jacobian criterion of cohomological smoothness for moduli spaces of sections of smooth algebraic varieties over the curve that allows us to solve our problems. (joint work with Peter Scholze) In our joint work with Scholze we need to give a meaning to statements like "the stack of principal G-bundles on the curve is smooth of dimension 0" and construct "smooth perfectoid charts on it". The problem is that in the perfectoid world there is no infinitesimals and thus no Jacobian criterion that would allow us to define what is a smooth morphism. The good notion in this setting is the one of a coho...

11F85 ; 11S31 ; 11R39 ; 14G22 ; 14H40

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Research talks;Algebra;Number Theory

We prove a finiteness result on the $p$-adic cohomology of the Lubin-Tate tower, which allows one to go from mod $p$ and $p$-adic
$GL_n (F)$-representations to Galois representations (compatibly with some global cor-respondences).

14G22 ; 22E50 ; 14F30

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Research talks;Number Theory

We report on progress towards a theory of local Shimura varieties for $p$-adic fields, in parallel with the theory of local shtukas for $\mathbb{F}_q((t))$. Using perfectoid spaces (in particular Scholze's theory of "diamonds"), it is possible to give a unified definition of local shtukas which incorporates both the equal and unequal characteristic cases. Conjecturally, if G is a reductive group, the cohomology of a moduli space of local G-shtukas should realize the Langlands correspondence for G in a systematic way (along the lines described by V. Lafforgue for global stukas). This talk will draw heavily from ideas of Peter Scholze and Laurent Fargues. We report on progress towards a theory of local Shimura varieties for $p$-adic fields, in parallel with the theory of local shtukas for $\mathbb{F}_q((t))$. Using perfectoid spaces (in particular Scholze's theory of "diamonds"), it is possible to give a unified definition of local shtukas which incorporates both the equal and unequal characteristic cases. Conjecturally, if G is a reductive group, the cohomology of a moduli space of local ...

14G35 ; 11S37

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- 331 p.
ISBN 978-2-85629-150-4

Astérisque , 0291

Localisation : Périodique 1er étage;Réserve

variété de Shimura # groupe p-divisible # espace de Rapoport-Zink # correspondance de Langlands # cohomologie étale des espaces rigides

11G18 ; 11Fxx ; 14G35 ; 14L05 ; 14G22

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- 406 p.
ISBN 978-3-7643-8455-5

Progress in mathematics , 0262

Localisation : Collection 1er étage

géométrie algébrique # théorie des nombres # groupe formels # géométrie analytique rigide # conjecture de Langland-Weil # variété de Shimura # groupes non-abéliens

14-02 ; 11-02 ; 14L05 ; 14G22 ; 11R39 ; 14G35

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- xiii; 382 p.
ISBN 978-2-85629-896-1

Astérisque , 0406

Localisation : Périodique 1er étage

théorie de Hodge $\rho$-adique # représentation galoisienne # fibré vectoriel # courbe algébrique

11G25 ; 11F80 ; 14G20 ; 14G22 ; 14H60

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