The Tate pairing for Abelian varieties over finite fields

Peter Bruin

Displaying similar documents to “The Tate pairing for Abelian varieties over finite fields”

The period-index problem in WC-groups IV: a local transition theorem

Pete L. Clark (2010)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a complete discretely valued field with perfect residue field $k$ . Assuming upper bounds on the relation between period and index for WC-groups over $k$ , we deduce corresponding upper bounds on the relation between period and index for WC-groups over $K$ . Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a “duality free” context. Our techniques include the use of of torsors under abelian varieties with good reduction and...

Equidistribution of small subvarieties of an Abelian variety.

Baker, Matthew, Ih, Su-ion (2004)

The New York Journal of Mathematics [electronic only]

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Explicit bounds for split reductions of simple abelian varieties

Jeffrey D. Achter (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $X / K$ be an absolutely simple abelian variety over a number field; we study whether the reductions $X_{𝔭}$ tend to be simple, too. We show that if $End (X)$ is a definite quaternion algebra, then the reduction $X_{𝔭}$ is geometrically isogenous to the self-product of an absolutely simple abelian variety for $𝔭$ in a set of positive density, while if $X$ is of Mumford type, then $X_{𝔭}$ is simple for almost all $𝔭$ . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit...

Integral canonical models of Shimura varieties

Mark Kisin (2009)

Journal de Théorie des Nombres de Bordeaux

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The aim of these notes is to provide an introduction to the subject of integral canonical models of Shimura varieties, and then to sketch a proof of the existence of such models for Shimura varieties of Hodge and, more generally, abelian type. For full details the reader is refered to [Ki 3].

Criterion for a field to be abelian

J. Wójcik (1995)

Colloquium Mathematicae

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Abelian varieties, surfaces of general type and integrable systems.

Lesfari, A. (2007)

Beiträge zur Algebra und Geometrie

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On a dynamical Brauer–Manin obstruction

Liang-Chung Hsia, Joseph Silverman (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $ϕ : X \to X$ be a morphism of a variety defined over a number field $K$ , let $V \subset X$ be a $K$ -subvariety, and let $𝒪_{ϕ} (P) = {ϕ^{n} (P) : n \geq 0}$ be the orbit of a point $P \in X (K)$ . We describe a local-global principle for the intersection $V \cap 𝒪_{ϕ} (P)$ . 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...