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Documents  Demarche, Cyril | enregistrements trouvés : 10

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- xiv; 292 p.
ISBN 978-2-85629-819-0

Panoramas et synthèses , 0046

Localisation : Collection 1er étage

algèbre à division # algèbre centrale simple # cohomologie non abélienne # cohomologie perverse # fibration de Hitchin # forme quadratique # gerbe # groupe classique # groupe linéaire # groupe orthogonal # lemme fondamental de Langlands-Shelstad # lemme fondamental pondéré d'Arthur # programme de Langlands # schéma en groupes # torseur

11F70 ; 11F72 ; 11R39 ; 14D20 ; 14L15 ; 14L30 ; 18F20 ; 20G07 ; 20G10 ; 20J06 ; 22E55

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Algebraic and Complex Geometry;Number Theory

Given a smooth scheme $X$ over the ring of integers of a $p$-adic field, we introduce the notion of a relative Breuil-Kisin-Fargues module $M$ on $X$. Each such $M$ simultaneously encodes the data of a lisse étale sheaf, a module with flat connection, and a crystal, whose cohomologies are then intertwined by a relative form of the $A_{inf}$ cohomology introduced in "Integral $p$-adic Hodge theory" by Bhatt-M-Scholze. They are moreover closely related to other work in relative $p$-adic Hodge theory, notably Faltings small generalised representations and his relative Fontaine Lafaille theory. Joint with Takeshi Tsuji. Given a smooth scheme $X$ over the ring of integers of a $p$-adic field, we introduce the notion of a relative Breuil-Kisin-Fargues module $M$ on $X$. Each such $M$ simultaneously encodes the data of a lisse étale sheaf, a module with flat connection, and a crystal, whose cohomologies are then intertwined by a relative form of the $A_{inf}$ cohomology introduced in "Integral $p$-adic Hodge theory" by Bhatt-M-Scholze. They are moreover closely ...

14F20 ; 14F30 ; 14F40 ; 14D10 ; 14G20 ; 14G22 ; 11G25

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Algebraic and Complex Geometry;Number Theory

We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their $L$-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be classical, the strategy consists in first realizing them as exponential motives and computing their Hodge numbers by means of the irregular Hodge filtration. We show that all Hodge numbers are either zero or one, which implies potential automorphicity thanks to recent results of Patrikis and Taylor. The first talk will be concerned with the arithmetic aspects and in the second one we will present the Hodge theoretic computations. Joint work with Claude Sabbah and Jeng-Daw Yu. We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their $L$-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be classical, the strategy consists in first realizing them as exponential motives and computing their Hodge numbers by means of the irregular ...

11L05 ; 14C30 ; 14F40 ; 32S40

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Algebraic and Complex Geometry

(joint work with G. Wüstholz) Roughly, $1$-dimensional periods are the complex numbers obtained by integrating a differential form on an algebraic curve over $\bar{\mathbf{Q}}$ over a suitable domain of integration. One of the alternative characterisations is as periods of Deligne $1$-motives.
We clear up the linear relations between these numbers, proving Kontsevich's version of the period conjecture for $1$-dimensional periods. In particular, a $1$-dimensional period is shown to be algebraic if and only if it is of the form $\int_\gamma (\phi+df)$ with $\int_\gamma\phi=0$. We also get formulas for the spaces of periods of a given $1$-motive, generalising Baker's theorem on logarithms of algebraic numbers.
The proof is based on a version of Wüstholz's analytic subgroup theorem for $1$-motives.
(joint work with G. Wüstholz) Roughly, $1$-dimensional periods are the complex numbers obtained by integrating a differential form on an algebraic curve over $\bar{\mathbf{Q}}$ over a suitable domain of integration. One of the alternative characterisations is as periods of Deligne $1$-motives.
We clear up the linear relations between these numbers, proving Kontsevich's version of the period conjecture for $1$-dimensional periods. In particular, ...

14F42 ; 19E15 ; 19F27

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Algebraic and Complex Geometry

​We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Although other geometric proofs of this result are known, our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow, and using them to quickly solve the problem by invoking the irreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs. Joint work with Davesh Maulik. ​We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Although other geometric proofs of this result are known, our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow, and using them to quickly solve the problem by invoking ...

14C15 ; 14J28 ; 32J27 ; 17B68

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Algebraic and Complex Geometry;Number Theory

11G18 ; 14F30 ; 11F75

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Algebraic and Complex Geometry;Number Theory

Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

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Algebraic and Complex Geometry;Number Theory

Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

... Lire [+]

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Algebraic and Complex Geometry;Number Theory

Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Algebraic and Complex Geometry;Number Theory

Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

... Lire [+]

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