Displaying similar documents to “Torsors under tori and Néron models”

Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

Marco Antei (2010)

Journal de Théorie des Nombres de Bordeaux

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We show that the natural morphism ϕ : π 1 ( X η , x η ) → π 1 ( X , x ) η between the fundamental group scheme of the generic fiber X η of a scheme X over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of X is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed G -torsor over X η to be extended over X . We finally provide examples where ϕ : π 1 ( X η , x η ) → π 1 ( X , x ) η is an isomorphism.

Examples of functions -extendable for each finite, but not ∞ -extendable

Wiesław Pawłucki (1998)

Banach Center Publications

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In Example 1, we describe a subset X of the plane and a function on X which has a k -extension to the whole ℝ 2 for each finite, but has no ∞ -extension to ℝ 2 . In Example 2, we construct a similar example of a subanalytic subset of ℝ 5 ; much more sophisticated than the first one. The dimensions given here are smallest possible.

On a dynamical Brauer–Manin obstruction

Liang-Chung Hsia, Joseph Silverman (2009)

Journal de Théorie des Nombres de Bordeaux

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Let ϕ : X → X be a morphism of a variety defined over a number field  K , let  V ⊂ X be a K -subvariety, and let  𝒪 ϕ ( P ) = { ϕ n ( P ) : n ≥ 0 } be the orbit of a point  P ∈ X ( K ) . We describe a local-global principle for the intersection  V ∩ 𝒪 ϕ ( P ) . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of  V ( K ) are Brauer–Manin unobstructed for power maps on  ℙ 2 in two cases: (1)  V is a translate of a torus. (2)  V is a line and  P has a preperiodic coordinate. A key tool in the proofs is the...

Conjugacy classes of series in positive characteristic and Witt vectors.

Sandrine Jean (2009)

Journal de Théorie des Nombres de Bordeaux

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Let k be the algebraic closure of 𝔽 p and K be the local field of formal power series with coefficients in k . The aim of this paper is the description of the set 𝒴 n of conjugacy classes of series of order p n for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic p which are invertible and of finite order p n for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series...

On the vanishing of Iwasawa invariants of absolutely abelian p-extensions

Gen Yamamoto (2000)

Acta Arithmetica

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1. Introduction. Let p be a prime number and ℤ p the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, k ∞ a ℤ p -extension of k, k n the nth layer of k ∞ / k , and A n the p-Sylow subgroup of the ideal class group of k n . Iwasawa proved the following well-known theorem about the order A n of A n : Theorem A (Iwasawa). Let k ∞ / k be a ℤ p -extension and A n the p-Sylow subgroup of the ideal class group of k n , where k n is the n th layer of k ∞ / k . Then there exist integers λ = λ ( k ∞ / k ) ≥ 0 , μ = μ ( k ∞ / k ) ≥ 0 , ν = ν ( k ∞ / k ) , and n₀ ≥ 0...

Best approximations and porous sets

Simeon Reich, Alexander J. Zaslavski (2003)

Commentationes Mathematicae Universitatis Carolinae

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Let D be a nonempty compact subset of a Banach space X and denote by S ( X ) the family of all nonempty bounded closed convex subsets of X . We endow S ( X ) with the Hausdorff metric and show that there exists a set ℱ ⊂ S ( X ) such that its complement S ( X ) ∖ ℱ is σ -porous and such that for each A ∈ ℱ and each x ˜ ∈ D , the set of solutions of the best approximation problem ∥ x ˜ - z ∥ → min , z ∈ A , is nonempty and compact, and each minimizing sequence has a convergent subsequence.