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2009•Journal de Théorie des Nombres de BordeauxOpen access

The Brauer–Manin obstruction for curves having split Jacobians

Samir Siksek

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Abstract

Let X → 𝒜 be a non-constant morphism from a curve X to an abelian variety 𝒜 , all defined over a number field k . Suppose that X is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on X to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that 𝒜 ( k ) and Ш ( 𝒜 / k ) are finite.

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Let X → 𝒜 be a non-constant morphism from a curve X to an abelian variety 𝒜 , all defined over a number field k . Suppose that X is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on X to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that 𝒜 ( k ) and Ш ( 𝒜 / k ) are finite.

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Available abstract

Let X → 𝒜 be a non-constant morphism from a curve X to an abelian variety 𝒜 , all defined over a number field k . Suppose that X is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on X to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that 𝒜 ( k ) and Ш ( 𝒜 / k ) are finite.

Key concepts: Hasse principle, Mathematics, Counterexample, Morphism, Abelian variety, Algebraic number field, Pure mathematics, Variety (cybernetics)

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