Displaying similar documents to “Explicit bounds for split reductions of simple abelian varieties”

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...

The Tate pairing for Abelian varieties over finite fields

Peter Bruin (2011)

Journal de Théorie des Nombres de Bordeaux

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In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey–Rück pairing, thereby giving a short proof of the perfectness of the latter.

Some remarks on almost rational torsion points

John Boxall, David Grant (2006)

Journal de Théorie des Nombres de Bordeaux

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For a commutative algebraic group G over a perfect field k , Ribet defined the set of almost rational torsion points G tors , k ar of G over k . For positive integers d , g , we show there is an integer U d , g such that for all tori T of dimension at most d over number fields of degree at most g , T tors , k ar ⊆ T [ U d , g ] . We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit...

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...