Groupes de Galois arithmétiques et différentiels by Daniel Bertrand, Pierre Dèbes

By Daniel Bertrand, Pierre Dèbes

Résumé :
Ce quantity constitue les actes du colloque sur les groupes de Galois arithmétiques et différentiels qui s'est déroulé au CIRM de Luminy (France) du eight au thirteen Mars 2004. Le yet était de rendre compte du rapprochement en cours entre les deux théories, et de le développer. Le quantity, à l'image du colloque, aborde des thèmes communs aux deux théories: espaces de modules (de courbes, de revêtements, de connexions), questions arithmétiques (corps de définition, théorie de los angeles descente), groupes fondamentaux, problèmes inverses, méthodes de déformation, calculs et réalisations explicites de groupes de Galois, points algorithmiques.

Mots clefs : Algorithmes, approximation, catégorie, catégorie des foncteurs, cohomologie parabolique, complexité algorithmique, connexions, correspondance de Riemann-Hilbert, correspondances, corps de fonctions, corps des éléments analytiques, courbes elliptiques, dessins d'enfants, diviseurs premiers de Zariski, D-modules locaux bornés, dualité de Poincaré, équations différentielles p-adiques, espaces de Hurwitz, fibré vectoriel, fonctions de Belyi, fonctions hypergéométriques, formes modulaires, Frattini, géométrie anabélienne, groupe de Galois différentiel, groupe fondamental, groupes linéaires algébriques sur les corps locaux et leurs anneaux de valuation, groupes de tresse, ID-modules (modules différentiels itératifs), inégalité de Bogomolov-Gieseker, irréductibilité, jacobienne, limite, computer de Turing, modules, monodromie, multiplicateurs de Schur, nombres p-adiques, opérateur différentiel, opérateurs de Lamé, Painlevé VI, issues rationnels, preuve formelle, problème de Galois inverse, problème de Riemann-Hilbert, réduction des ID-modules, représentation de monodromie, représentation modulaire, représentations, revêtement des courbes, revêtement universel, ideas algébriques, stabilité, système différentiel fuchsien, temps polynomial déterministe, théorie de décomposition de Hilbert, théorie de Galois, théorie de Galois différentielle, théorie de Galois inverse, théorie de Galois pro-$\ell $, excursions modulaires, uniformisation, variété algébrique,

Abstract:
Arithmetic and differential Galois groups
On March 8-13, 2004, a gathering was once geared up on the Luminy CIRM (France) on mathematics and differential Galois teams, reflecting the transforming into interactions among the 2 theories. the current quantity collects the court cases of this convention. It covers the next subject matters: moduli areas (of curves, of coverings, of connexions), together with the new advancements on modular towers; the mathematics of coverings and of differential equations (fields of definition, descent theory); primary teams; the inverse difficulties and strategies of deformation; and the algorithmic elements of the theories, with particular computations or realizations of Galois groups.

Key phrases: Algebraic recommendations, algebraic type, algorithmic complexity, algorithms, anabelian geometry, Belyi services, Bogomolov-Gieseker inequality, braid teams, braid workforce and Hurwitz monodromy workforce, classification, complicated approximation, connections, correspondences, covers of curves, dessins d'enfants, deterministic polynomial time, differential Galois crew, differential Galois thought, differential operator, elliptic curves, fields of analytic parts, formalized facts, Frattini, Frattini and Spin covers, functionality fields, functor classification, basic team, Fuchsian differential platforms Galois thought, Hilbert decomposition conception, Hurwitz areas, hypergeometric capabilities ID-modules (iterative differential modules), inverse challenge of Galois concept, irreducibility, jacobian kind, j-line covers, Lamé differential operators, restrict, linear algebraic teams over neighborhood fields and their integers, in the community bounded D-modules, modular varieties, modular illustration, modular towers, moduli, moduli areas of covers, monodromy, monodromy illustration, p-adic differential equations, p-adic numbers, Painlevé VI, parabolic cohomology, pro-$\ell $ Galois idea, Poincaré duality, rational issues, aid of ID-modules, representations, Riemann-Hilbertcorrespondence, Riemann-Hilbert challenge, Serre's lifting invariant, Schur multiplier, balance, Turing computing device, uniformization, common hide, valuations, vector bundles, Zariski major divisors,

Class. math. : 03B35, 11F11, 11F25, 11F30, 11F32, 11Gxx, 11G18, 11R58, 11Y16, 11Y35, 12E, 12E30, 12F, 12F10, 12F12, 12G, 12G99, 12H05, 12H25, 12J, 13N, 13N05, 13N10, 14-04, 14D, 14Dxx, 14D22, 14F05, 14G05, 14G32, 14G35, 14H05, 14H10, 14H30, 18A25, 20B05, 20C05, 20C20, 20C25, 20D25, 20E18, 20E22, 20F34, 20F69, 20G, 20G25, 20J05, 20J06, 32J25, 32S40, 33C05, 34xx, 34A20, 34M55, 35C10, 35C20, 53G, 65E05, 65Y20, 68Q15

Table of Contents

* M. Berkenbosch -- Algorithms and moduli areas for differential equations
* M. Berkenbosch and M. van der placed -- households of linear differential equations at the projective line
* P. Boalch -- short creation to Painlevé VI
* A. Buium -- Correspondences, Fermat quotients, and uniformization
* J.-M. Couveignes -- Jacobiens, jacobiennes et stabilité numérique
* P. Débes -- An advent to the modular tower program
* M. Dettweiler and S. Wewers -- version of parabolic cohomology and Poincaré duality
* M. D. Fried -- the most conjecture of modular towers and its larger rank generalization
* R. Liţcanu and L. Zapponi -- homes of Lamé operators with finite monodromy
* S. Malek -- at the Riemann-Hilbert challenge and good vector bundles at the Riemann sphere
* B. H. Matzat -- quintessential p-adic differential modules
* F. Pop -- Galois conception of Zariski major divisors
* M. Romagny and S. Wewers -- Hurwitz spaces
* D. Semmen -- the crowd idea at the back of modular towers
* C. Simpson -- Formalized facts, computation, and the development challenge in algebraic geometry
* Annexe. Liste des members

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Extra resources for Groupes de Galois arithmétiques et différentiels

Example text

Let KZ be the Picard-Vessiot extension of LZ . By the definition of a standard operator we can Z(G) write KZ = C(Z). Let VZ ⊂ KZ be the solution space of LZ . In the compositum ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 2006 M. BERKENBOSCH 32 K of K and KZ over C(Z), we have the identity VZ = f V , for the appropriate f, as defined in the construction of LZ . 7.

T = NA2 + NB1 . Now the local exponents at some point We can make a decomposition N 2 p ∈ k¯ are the solutions of the equation λ(λ−1) = λ satisfy λ(λ − 1) = A(p) N2 (p)2 . A·(x−p)2 |x=p . N22 So the local exponents For the D2 -case we search for points with local exponent difference in 12 +Z. Because 3−2n L is in normal form, the local exponents of L at such a point are 2n+1 , for 4 , 4 some n ∈ Z. Therefore we get the system of equations: D2 -case : (3 + 4n − 4n2 )N2 (p)2 + 16A(p) = 0 N2 (p) = 0.

Then φF = φf ◦ φh , and we only need to show that f is Gal(k/k) −1 invariant. But we have that φσ(f ) = φσ(F ) ◦ φ−1 = φf σ(h) = φF ◦ φS(σ) ◦ (φh ◦ φS(σ) ) and therefore f ∈ k(x). 14. — The above corollary states that every differential operator ∂x2 − r, with r ∈ k(x) is the pullback of a differential operator over k(x) with three singularities, and with the same local exponents as the corresponding standard operator (use Norm(φh (StG ))). So we can see this corollary as a “rational version” of Klein’s theorem.

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