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# On the Notion of Visibility of Torsors

Published:2011-10-05
Printed: Jun 2013
• Amod Agashe,
Department of Mathematics, Florida State University, Tallahassee, FL, U.S.A.
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## Abstract

Let $J$ be an abelian variety and $A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$. Let $x$ be an element of $H^1(\mathbf{Q},A)$. Then there are at least two definitions of $x$ being visible in $J$: one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$ to a subvariety of $J$, and the other asks that $x$ be in the kernel of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we clarify the relation between the two definitions.
 Keywords: torsors, principal homogeneous spaces, visibility, Shafarevich-Tate group
 MSC Classifications: 11G35 - Varieties over global fields [See also 14G25] 14G25 - Global ground fields