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𝒟 -modules arithmétiques surcohérents. Application aux fonctions L

Daniel Caro (2004)

Annales de l’institut Fourier

Nous étudions d’abord le foncteur cohomologique local. Ensuite, nous introduisons la notion de 𝒟 -modules arithmétiques surcohérents. Nous prouvons que les F - isocristaux unités sont surcohérents et surtout que la surcohérence est stable par images directes, images inverses extraordinaires et foncteurs cohomologiques locaux. On obtient, via cette stabilité, une formule cohomologique pour les fonctions L associées aux complexes duaux de complexes surcohérents. Celle-ci étend celle d’Étesse et Le Stum...

2-Cohomology of semi-simple simply connected group-schemes over curves defined over p -adic fields

Jean-Claude Douai (2013)

Journal de Théorie des Nombres de Bordeaux

Let X be a proper, smooth, geometrically connected curve over a p -adic field k . Lichtenbaum proved that there exists a perfect duality: Br ( X ) × Pic ( X ) → ℚ / ℤ between the Brauer and the Picard group of X , from which he deduced the existence of an injection of Br ( X ) in ∏ P ∈ X Br ( k P ) where P ∈ X and k P denotes the residual field of the point P . The aim of this paper is to prove that if G = G ˜ is an X e t - scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results the neutrality of every X e t -gerb which...

A Bogomolov property for curves modulo algebraic subgroups

Philipp Habegger (2009)

Bulletin de la Société Mathématique de France

Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least 2 . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

A computer algebra solution to a problem in finite groups.

Gert-Martin Greuel (2003)

Revista Matemática Iberoamericana

We report on a partial solution of the conjecture that the class of finite solvable groups can be characterised by 2-variable identities. The proof requires pieces from number theory, algebraic geometry, singularity theory and computer algebra. The computations were carried out using the computer algebra system SINGULAR.

A descent map for curves with totally degenerate semi-stable reduction

Shahed Sharif (2013)

Journal de Théorie des Nombres de Bordeaux

Let K be a local field of residue characteristic p . Let C be a curve over K whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to- p rational torsion subgroup on the Jacobian of C . We also determine divisibility of line bundles on C , including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of C .

A dimension formula for Ekedahl-Oort strata

Ben Moonen (2004)

Annales de l’institut Fourier

We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by the classes in a Weyl group modulo a subgroup, and each class has a distinguished representative of minimal length. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.

A dynamical Shafarevich theorem for twists of rational morphisms

Brian Justin Stout (2014)

Acta Arithmetica

Let K denote a number field, S a finite set of places of K, and ϕ: ℙⁿ → ℙⁿ a rational morphism defined over K. The main result of this paper states that there are only finitely many twists of ϕ defined over K which have good reduction at all places outside S. This answers a question of Silverman in the affirmative.

A family of varieties with exactly one pointless rational fiber

Bianca Viray (2010)

Journal de Théorie des Nombres de Bordeaux

We construct a concrete example of a 1 -parameter family of smooth projective geometrically integral varieties over an open subscheme of ℙ ℚ 1 such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.

A finiteness result for the compactly supported cohomology of rigid analytic varieties, II

Roland Huber (2007)

Annales de l’institut Fourier

Let h : X → Y be a separated morphism of adic spaces of finite type over a non-archimedean field k with Y affinoid and of dimension ≤ 1 , let L be a locally closed constructible subset of X and let g : ( X , L ) → Y be the morphism of pseudo-adic spaces induced by h . Let A be a noetherian torsion ring with torsion prime to the characteristic of the residue field of the valuation ring of k and let F be a constant A -module of finite type on ( X , L ) e ´ t . There is a natural class 𝒞 ( Y ) of A -modules on Y e ´ t generated by the constructible A -modules...

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